**Use of Fast Fourier Transform for Sensitivity Analysis**

Andrej Prošek and Matjaž Leskovar

Additional information is available at the end of the chapter

http://dx.doi.org/10.5772/59769

## **1. Introduction**

The uncertainty quantification of code calculations is typically accompanied by a sensitivity analysis, in which the influence of the individual contributors to the uncertainty is determined. In the sensitivity analysis, the basic step is to perform sensitivity calculations varying the input parameters. One or more input parameters could be varied at a time. The typical statistical methods for the sensitivity analysis used in uncertainty methods are for example Pearson's Correlation Coefficient, Standardized Regression Coefficient, Partial Correlation Coefficient and others [1]. The output results are time domain signals. The objective of this study was to use fast Fourier transform (FFT) based approaches to determine the sensitivity of output parameters. In the reference [2], the FFT based approaches have been used for the accuracy quantification. The difference between the accuracy quantification and the sensitivity analysis is that in the accuracy quantification the experimental data are compared to the code calculated data, while in the sensitivity analysis the reference calculation signal is compared to the sensitivity run calculation signal. To do this comparison, first the fast Fourier transform is used to transform time domain signals into frequency domain signals. Then, the average amplitude is calculated, which is the sum of the amplitudes of the frequency domain difference signal (between the sensitivity run calculation signal and the reference calculation signal) normalized by the sum of the amplitudes of the frequency domain reference signal. Finally, the figures of merit based on the average amplitude are used to judge the sensitivity.

Such a FFT based approach is different from the typical sensitivity analysis using a statistical procedure to determine the influence of sensitive input parameters on the output parameter. Namely, the influence of the sensitive input parameters is represented by the average ampli‐ tude which remembers the previous history.

In this study it is shown that the proposed FFT based tool is complementary and a good alternative to the mentioned typical statistical methods, if one parameter is varied at a time.

© 2015 The Author(s). Licensee InTech. This chapter is distributed under the terms of the Creative Commons Attribution License (http://creativecommons.org/licenses/by/3.0), which permits unrestricted use, distribution, and eproduction in any medium, provided the original work is properly cited.

The advantage of the proposed method is that the results of the sensitivity analysis obtained by the FFT based tool could be ranked. It provides a consistent ranking of sensitive input parameters according to their influence to the output parameter when one parameter is varied at a time, and based on the fact that the same method can be used for more participants performing calculations. For example, there is no need to have ranking levels like it was proposed in the Phase III of the Best Estimate Methods – Uncertainty and Sensitivity Evalua‐ tion (BEMUSE) programme [3] for the qualitative judgement, because in this approach the figure of merit is a quantitative value and therefore can be directly ranked. The zero value of the figure of merit means a not relevant sensitive parameter and the larger the figure of merit is the more influential the input parameter is.

The difference between statistical methods and the FFT based approach is that in the case of statistical methods the influence of each varied input parameter on the output result can be obtained even if more parameters are varied at a time. This cannot be done by FFT based tools when more parameters are varied at a time. Rather, the total influence of sensitive parameters on the result is given by the FFT based tool. But the good thing is that the same measure is used for both single and multiple variations and in this way the compensation effects of the influence of different sensitive parameters could be studied. As was already mentioned the FFT based approaches are complementary to statistical methods.

In this Chapter, first the original fast Fourier transform based method (FFTBM) approach is described. The average amplitude, the signal mirroring, the FFTBM Add-in tool and the time dependent accuracy measures are introduced. Based on signal mirroring the improved FFTBM by signal mirroring (FFTBM-SM) was developed. By calculating the time dependent average amplitude it can be answered, which discrepancy due to the parameter variation contributes to the sensitivity and how much is its contribution. The past application of the FFT based tool for the accuracy quantification showed that the original FFTBM gave an unrealistic judgment of the average amplitude for monotonically increasing or decreasing functions, causing problems in the FFTBM results interpretation. It was found out [4] that the reason for such an unrealistic calculated accuracy of increasing/decreasing signals is the edge between the first and last data point of the investigated signal, when the signal is periodically extended. Namely, if the values of the first and last data point of the investigated signal differ, then there are discontinuities present in the periodically extended signal seen by the discrete Fourier transform, which views the finite domain signal as an infinite periodic signal. The disconti‐ nuities give several harmonic components in the frequency domain, thus increasing the sum of the amplitudes, on which FFTBM is based, and by this influencing the accuracy. The influence of the edge due to the periodically extended signal is for clarity reasons called the edge effect.

Then the methods used for the sensitivity analysis are described. For the demonstration of the sensitivity study using FFT based tools the L2-5 test, which simulates the large break loss of coolant accident in the Loss of Fluid test (LOFT) facility, was used. The signals used were obtained from the Organisation for Economic Co-operation and Development (OECD) BEMUSE project. In the BEMUSE project there were 14 participants, each performing a reference calculation and 15 sensitivity runs of the LOFT L2-5 test. Three output parameters were provided: the upper plenum pressure, the primary side mass inventory and the rod surface temperature.

Finally, the application of the FFT based approaches to the sensitivity analysis is described. Both FFTBM and FFTBM-SM were used.

## **2. Fast fourier transform based method description**

The advantage of the proposed method is that the results of the sensitivity analysis obtained by the FFT based tool could be ranked. It provides a consistent ranking of sensitive input parameters according to their influence to the output parameter when one parameter is varied at a time, and based on the fact that the same method can be used for more participants performing calculations. For example, there is no need to have ranking levels like it was proposed in the Phase III of the Best Estimate Methods – Uncertainty and Sensitivity Evalua‐ tion (BEMUSE) programme [3] for the qualitative judgement, because in this approach the figure of merit is a quantitative value and therefore can be directly ranked. The zero value of the figure of merit means a not relevant sensitive parameter and the larger the figure of merit

The difference between statistical methods and the FFT based approach is that in the case of statistical methods the influence of each varied input parameter on the output result can be obtained even if more parameters are varied at a time. This cannot be done by FFT based tools when more parameters are varied at a time. Rather, the total influence of sensitive parameters on the result is given by the FFT based tool. But the good thing is that the same measure is used for both single and multiple variations and in this way the compensation effects of the influence of different sensitive parameters could be studied. As was already mentioned the

In this Chapter, first the original fast Fourier transform based method (FFTBM) approach is described. The average amplitude, the signal mirroring, the FFTBM Add-in tool and the time dependent accuracy measures are introduced. Based on signal mirroring the improved FFTBM by signal mirroring (FFTBM-SM) was developed. By calculating the time dependent average amplitude it can be answered, which discrepancy due to the parameter variation contributes to the sensitivity and how much is its contribution. The past application of the FFT based tool for the accuracy quantification showed that the original FFTBM gave an unrealistic judgment of the average amplitude for monotonically increasing or decreasing functions, causing problems in the FFTBM results interpretation. It was found out [4] that the reason for such an unrealistic calculated accuracy of increasing/decreasing signals is the edge between the first and last data point of the investigated signal, when the signal is periodically extended. Namely, if the values of the first and last data point of the investigated signal differ, then there are discontinuities present in the periodically extended signal seen by the discrete Fourier transform, which views the finite domain signal as an infinite periodic signal. The disconti‐ nuities give several harmonic components in the frequency domain, thus increasing the sum of the amplitudes, on which FFTBM is based, and by this influencing the accuracy. The influence of the edge due to the periodically extended signal is for clarity reasons called the

Then the methods used for the sensitivity analysis are described. For the demonstration of the sensitivity study using FFT based tools the L2-5 test, which simulates the large break loss of coolant accident in the Loss of Fluid test (LOFT) facility, was used. The signals used were obtained from the Organisation for Economic Co-operation and Development (OECD) BEMUSE project. In the BEMUSE project there were 14 participants, each performing a reference calculation and 15 sensitivity runs of the LOFT L2-5 test. Three output parameters

is the more influential the input parameter is.

52 Fourier Transform - Signal Processing and Physical Sciences

edge effect.

FFT based approaches are complementary to statistical methods.

The FFT based method is proposed for the sensitivity analysis, which is analogous to the FFT based code-accuracy assessment described in ref. [5]. The FFT based approach for code accuracy consists of three steps: a) selection of the test case (experimental or plant measured time trends to compare), b) qualitative analysis, and c) quantitative analysis. The qualitative analysis is necessary before quantifying the discrepancies between the measured and calcu‐ lated trends. The qualitative analysis includes also the visual observation of plots and the evaluation of the discrepancies between the measured and calculated trends, which should be predictable and understood. For the sensitivity analysis the same FFT based approach is used for the quantitative analysis as used for the code-accuracy. However, the signals compared now are the output signal obtained with the reference value of the input parameter and the output signal as the result of the sensitive input parameter variation.

In the quantitative analysis, the influence of the sensitive input parameter variation is judged in the frequency domain. Therefore the time domain signals used in the sensitivity analysis have to be transformed in the frequency domain signals. The addressed time domain signals assume values different from zero only in the interval [0, Td], where Td is the duration of the signal. Also the digital computers can only work with information that is discrete and finite in length (e.g. *N* points) and there is no version of the Fourier transform that uses finite length signals [6]. The way around this is to make the finite data look like an infinite length signal. This is done by imagining that the signal has an infinite number of samples on the left and right of the actual points. The imagined samples can be a duplication of the actual data points. In this case, the signal looks discrete and periodic. This calls for the discrete Fourier transform (DFT) to be used. There are several ways to calculate DFT. One method is FFT. While it produces the same results as the other approaches, it is incredibly more efficient. The key point to understand the FFTBM is that the periodicity is invoked in order to be able to use a mathematical tool, i.e., the DFT. It seems that the developers of the original FFTBM have not been sufficiently aware of this fact.

The discrete Fourier transform views both, the time domain and the frequency domain, as periodic [6]. However, the signals to be used for the comparison are not periodic and the user must conform to the DFT's view of the world. When a new period starts, the *N* samples on the left side are not related to the samples on the right side. However, DFT views these *N* points to be a single period of an infinitely long periodic signal. This means that the left side of the signal is connected to the right side of the signal, and vice versa. The most serious consequence of the time domain periodicity is the occurrence of the edge, where the signals are glued. When the signal spectrum is calculated with DFT, the edge is taken into account, despite the fact that the edge has no physical meaning for the comparison, since it was introduced artificially by the applied numerical method. It is known that the edge produces a variegated spectrum of frequencies due to the discontinuity of the edge. These frequencies originating from the artificially introduced edge may overshadow the frequency spectrum of the investigated signal. Therefore an improved version of FFTBM by signal mirroring has been proposed, which is described in detail in [4]. Both the original FFTBM and the improved FFTBM by signal mirroring have been used in the demonstration application. The same equations are used for the calculation of the average amplitude, like for the original FFTBM, except that, instead of the original signals, the symmetrized signals are used (for further details see ref. [4]). In the following it is first described how the average amplitude is calculated.

#### **2.1. Average amplitude**

FFT is another method for calculating the DFT. While it produces the same result as the other approaches, it significantly reduces the computation time. FFT usually operates with a number of values *N* that is a power of two. Typically, *N* is selected between 32 and 4096 [6]. In addition, the sampling theorem must be fulfilled to avoid the distortion of sampled signals due to the aliasing occurrence. The sampling theorem says: "a signal that varies continuously with time is completely determined by its values at an infinite sequence of equally spaced times if the frequency of these sampling times is greater than twice the highest frequency component of the signal" [7]. Thus if the number of points defining the function in the time domain is *N*=2m +1, then according to the sampling theorem the sampling frequency is:

$$\frac{1}{\tau} = f\_s = \text{2} \\ f\_{\text{max}} = \frac{N}{t\_d} = \frac{\text{2}^{m+1}}{t\_d} \tag{1}$$

where *τ* is the sampling interval, *t*d is the transient time duration of the sampled signal and *f*max is the highest (maximum) frequency component of the signal. The sampling theorem does not hold beyond *f*max. From the relation in Eq. (1) it is seen that the number of points selection is strictly connected to the sampling frequency. The FFT algorithm requires the number of points, equally spaced, which is a power with base 2. Generally an interpolation is necessary to satisfy this requirement. The original FFTBM is done so that the default value of the exponent m ranges from 8 to 11. This gives *N* ranging from 512 to 4096. The final number of points used by FFTBM is determined depending on the value of *f*fix, which is the minimum requested maximum frequency and is input value. If *f*max is not larger than *f*fix, the number of points is doubled (exponent m is increased for 1) until the criterion is satisfied or the exponent m equals to 11. Please note, that the minimum value of the exponent m is 8. The FFTBM application implies the following input values: the fixed frequency *f*fix (minimum maximum frequency of the analysis, this determines the number of points *N*), the cut off frequency (*f*cut), the start time *t*<sup>s</sup> and the end time *t*<sup>e</sup> of the analysed window (determines the analysis window *t*d = *t*s - *t*e). A cut off frequency has been introduced to cut off spurious contributions, generally negligible. When *f*cut is equal or larger than *f*fix, all frequency components are considered.

For the calculation of the differences between the output signal obtained with the reference value of the input parameter (reference signal *F*ref*(t)*) and the output signal as the result of the sensitive input parameter variation (sensitive signal *F*sen*(t)*), the reference signal (*F*ref*(t)*) and the difference signal *ΔF*(*t*) are needed. The difference signal in the time domain is defined as:

$$
\Delta F(t) = F\_{\nu f}(t) - F\_{\text{sen}}(t). \tag{2}
$$

After performing the fast Fourier transform the obtained spectra of amplitudes are used for the calculation of the average amplitude (*AA*):

$$\mathbf{AA} = \frac{\sum\_{n=0}^{2^{\text{nr}}} |\tilde{\Delta}F(f\_n)|}{\sum\_{n=0}^{2^{\text{nr}}} |\tilde{F}\_{\text{ref}}(f\_n)|},\tag{3}$$

where |*Δ*˜*<sup>F</sup>* ( *<sup>f</sup> <sup>n</sup>*)| is the difference signal amplitude at frequency *f*n and | *<sup>F</sup>*˜ ref( *f <sup>n</sup>*)| is the reference signal amplitude at frequency *f*n. The AA factor can be considered as a sort of average fractional difference and the closer the AA value is to zero, the smaller is the sensitivity (influence). In our specific application, the larger the sensitivity is the larger is the difference between the signals, normally resulting in a larger AA value. Typically, based on the previous experience in the accuracy quantification the values of AA below 0.3 indicate a small influence (in the case of pressure below 0.1), while the values above 0.5 indicate a large influence.

The above Eq. (3) can be also viewed as:

the edge has no physical meaning for the comparison, since it was introduced artificially by the applied numerical method. It is known that the edge produces a variegated spectrum of frequencies due to the discontinuity of the edge. These frequencies originating from the artificially introduced edge may overshadow the frequency spectrum of the investigated signal. Therefore an improved version of FFTBM by signal mirroring has been proposed, which is described in detail in [4]. Both the original FFTBM and the improved FFTBM by signal mirroring have been used in the demonstration application. The same equations are used for the calculation of the average amplitude, like for the original FFTBM, except that, instead of the original signals, the symmetrized signals are used (for further details see ref. [4]). In the

FFT is another method for calculating the DFT. While it produces the same result as the other approaches, it significantly reduces the computation time. FFT usually operates with a number of values *N* that is a power of two. Typically, *N* is selected between 32 and 4096 [6]. In addition, the sampling theorem must be fulfilled to avoid the distortion of sampled signals due to the aliasing occurrence. The sampling theorem says: "a signal that varies continuously with time is completely determined by its values at an infinite sequence of equally spaced times if the frequency of these sampling times is greater than twice the highest frequency component of the signal" [7]. Thus if the number of points defining the function in the time domain is *N*=2m

1

== = = (1)

+

*m*

*d d*

where *τ* is the sampling interval, *t*d is the transient time duration of the sampled signal and *f*max is the highest (maximum) frequency component of the signal. The sampling theorem does not hold beyond *f*max. From the relation in Eq. (1) it is seen that the number of points selection is strictly connected to the sampling frequency. The FFT algorithm requires the number of points, equally spaced, which is a power with base 2. Generally an interpolation is necessary to satisfy this requirement. The original FFTBM is done so that the default value of the exponent m ranges from 8 to 11. This gives *N* ranging from 512 to 4096. The final number of points used by FFTBM is determined depending on the value of *f*fix, which is the minimum requested maximum frequency and is input value. If *f*max is not larger than *f*fix, the number of points is doubled (exponent m is increased for 1) until the criterion is satisfied or the exponent m equals to 11. Please note, that the minimum value of the exponent m is 8. The FFTBM application implies the following input values: the fixed frequency *f*fix (minimum maximum frequency of the analysis, this determines the number of points *N*), the cut off frequency (*f*cut), the start time *t*<sup>s</sup> and the end time *t*<sup>e</sup> of the analysed window (determines the analysis window *t*d = *t*s - *t*e). A cut off frequency has been introduced to cut off spurious contributions, generally negligible.

following it is first described how the average amplitude is calculated.

+1, then according to the sampling theorem the sampling frequency is:

*s*

t

max 1 2 <sup>2</sup>

*t t*

When *f*cut is equal or larger than *f*fix, all frequency components are considered.

*<sup>N</sup> f f*

**2.1. Average amplitude**

54 Fourier Transform - Signal Processing and Physical Sciences

$$\mathbf{AA} = \frac{\mathbf{AA}\_{\text{dif}}}{\mathbf{AA}\_{\text{mf}}},$$

where AAdif is the average amplitude of the difference signal and AAref is the average ampli‐ tude of the reference signal:

$$\mathbf{AA}\_{\mathrm{dif}} = \frac{1}{2^m + 1} \sum\_{n=0}^{2^m} \left| \tilde{\Delta} F(f\_n) \right| \tag{5}$$

$$\mathbf{AA}\_{\text{ref}} = \frac{1}{2^{\text{m}} + 1} \sum\_{n=0}^{2^{\text{m}}} \left| \tilde{F}\_{\text{ref}}(f\_n) \right|. \tag{6}$$

If the reference and sensitive signals are the same, the difference signal is zero. The larger the difference is, the larger is AAdif (in principle). On the other hand, AAref normalizes the aver‐ age amplitude AA and the higher the sum of amplitudes is, the lower is the average ampli‐ tude AA. This means that for ranking the sensitivities of the selected output parameter only AAdifhasaninfluence.Ontheotherhand,forjudgingtheinfluenceofasinglesensitiveparameter variation on different output parameters, besides AAdif also AAref influences the ranking due to the different AAref the output parameters have.

$$\begin{aligned} \text{fraction } A0 &= \frac{\left| \tilde{\Delta} F(f\_0) \right|}{2^m}, \\ &\sum\_{n=0}^m \left| \tilde{\Delta} F(f\_n) \right| \end{aligned} \tag{7}$$

where <sup>|</sup>*<sup>Δ</sup>*˜*<sup>F</sup>* ( *<sup>f</sup>* 0)| is the mean value of the difference signal and it is equivalent to the mean error (ME)inthe timedomainasdefinedinref.[8].Themeasure *fractionA0* showswhenthe frequency amplitudes are dominating or when the mean values (like constant differences) are dominat‐ ing the sum of the amplitudes.

#### **2.2. Signal mirroring**

Ifwehaveafunction*F* (*t*) where 0≤*t* ≤*t<sup>d</sup>* and*t<sup>d</sup>* is thetransienttimeduration,itsmirroredfunction is defined as *Fmir*(*t*)= *F* (−*t*), where −*t<sup>d</sup>* ≤*t* ≤0. From these functions a new function is composed which is symmetrical in regard to the y-axis: *Fm*(*t*), where −*t<sup>d</sup>* ≤*t* ≤*t<sup>d</sup>* . This is illustrated in Figure 1. By composing the original signal (shown in Figure 1(a)) and its mirrored signal (signal mirroring), a signal without the edge between the first and the last data sample is obtained, and it is called symmetrized signal (shown in Figure 1(b)). It has the double number of points in order not to lose any information. Also it should be noted that the edge is not present in the original time domain signal (see Figure 1(a)). However, when performing FFT, the aperiodic original signalis treatedas aperiodic original signal asmentionedbefore andtherefore the edge is part of the periodic original signal, what is not physical. In the case of the symmetrized signal the edge is not present even when treating the signal as periodic.

For the calculation of the average amplitude by signal mirroring AAm the Eq. (3) is used like for the calculation of AA, except that, instead of the original signals, the symmetrized signals are used. This may be efficiently done by signal mirroring, where the investigated signal is mirrored before the original FFTBM is applied. By composing the original signal and its mirrored signal (signal mirroring), a symmetric signal (also called symmetrized signal) with the same characteristics is obtained, but without introducing the edge when viewed as an infinite periodic signal (for details refer to refs. [4, 9]). FFTBM using the symmetrized signals is called FFTBM-SM.

7

1 physical. In the case of the symmetrized signal the edge is not present even when treating

3 For the calculation of the average amplitude by signal mirroring AAm the Eq. (3) is used like 4 for the calculation of AA, except that, instead of the original signals, the symmetrized 5 signals are used. This may be efficiently done by signal mirroring, where the investigated 6 signal is mirrored before the original FFTBM is applied. By composing the original signal

11 **Fig.1.** Examples of (a) original digital signal and (b) symmetrized digital signal **Figure 1.** Examples of (a) original digital signal and (b) symmetrized digital signal

Use of Fast Fourier Transform for Sensitivity Analysis

#### 13 To take the advantages of spread sheets in preparing input forms, analysing data (including **2.3. FFTBM Add-in tool**

12 **2.3. FFTBM Add-in tool** 

10 symmetrized signals is called FFTBM-SM.

2 the signal as periodic.

If the reference and sensitive signals are the same, the difference signal is zero. The larger the difference is, the larger is AAdif (in principle). On the other hand, AAref normalizes the aver‐ age amplitude AA and the higher the sum of amplitudes is, the lower is the average ampli‐ tude AA. This means that for ranking the sensitivities of the selected output parameter only AAdifhasaninfluence.Ontheotherhand,forjudgingtheinfluenceofasinglesensitiveparameter variation on different output parameters, besides AAdif also AAref influences the ranking due

0

*ΔF(f )*

*n*

% (7)

2

*ΔF(f ) fraction A*

=

*n*

0

where <sup>|</sup>*<sup>Δ</sup>*˜*<sup>F</sup>* ( *<sup>f</sup>* 0)| is the mean value of the difference signal and it is equivalent to the mean error (ME)inthe timedomainasdefinedinref.[8].Themeasure *fractionA0* showswhenthe frequency amplitudes are dominating or when the mean values (like constant differences) are dominat‐

Ifwehaveafunction*F* (*t*) where 0≤*t* ≤*t<sup>d</sup>* and*t<sup>d</sup>* is thetransienttimeduration,itsmirroredfunction is defined as *Fmir*(*t*)= *F* (−*t*), where −*t<sup>d</sup>* ≤*t* ≤0. From these functions a new function is composed which is symmetrical in regard to the y-axis: *Fm*(*t*), where −*t<sup>d</sup>* ≤*t* ≤*t<sup>d</sup>* . This is illustrated in Figure 1. By composing the original signal (shown in Figure 1(a)) and its mirrored signal (signal mirroring), a signal without the edge between the first and the last data sample is obtained, and it is called symmetrized signal (shown in Figure 1(b)). It has the double number of points in order not to lose any information. Also it should be noted that the edge is not present in the original time domain signal (see Figure 1(a)). However, when performing FFT, the aperiodic original signalis treatedas aperiodic original signal asmentionedbefore andtherefore the edge is part of the periodic original signal, what is not physical. In the case of the symmetrized signal

For the calculation of the average amplitude by signal mirroring AAm the Eq. (3) is used like for the calculation of AA, except that, instead of the original signals, the symmetrized signals are used. This may be efficiently done by signal mirroring, where the investigated signal is mirrored before the original FFTBM is applied. By composing the original signal and its mirrored signal (signal mirroring), a symmetric signal (also called symmetrized signal) with the same characteristics is obtained, but without introducing the edge when viewed as an infinite periodic signal (for details refer to refs. [4, 9]). FFTBM using the symmetrized signals

the edge is not present even when treating the signal as periodic.

=

å

0 , *<sup>m</sup>*

%

to the different AAref the output parameters have.

56 Fourier Transform - Signal Processing and Physical Sciences

ing the sum of the amplitudes.

**2.2. Signal mirroring**

is called FFTBM-SM.

14 analysis of values), modifying graphs and the capability to store time recorded data, plots, 15 input forms and results, the Jožef Stefan Institute (JSI) in-house Microsoft Excel Add-in for 16 the accuracy evaluation of thermal-hydraulic code calculations with FFTBM has been 17 developed in 2003 [10]. Later the tool was upgraded with the capability to symmetrize the 18 signals, and some other improvements. The upgraded tool is called JSI FFTBM Add-in 2007 19 [11] and it has been used for the sensitivity study, described in this chapter. It includes both 20 FFTBM and FFTBM-SM. As already mentioned, the difference between FFTBM and FFTBM-21 SM is that in the latter the signals are symmetrized to eliminate the edge effect in calculating 22 the average amplitude by signal mirroring (AAm). 23 JSI FFTBM Add-in 2007 provides additional information on interpolated data of the signals 24 used, the difference signals, the amplitude spectra and the AA dependency on the cut To take the advantages of spread sheets in preparing input forms, analysing data (including analysis of values), modifying graphs and the capability to store time recorded data, plots, input forms and results, the Jožef Stefan Institute (JSI) in-house Microsoft Excel Add-in for the accuracy evaluation of thermal-hydraulic code calculations with FFTBM has been developed in 2003 [10]. Later the tool was upgraded with the capability to symmetrize the signals, and some other improvements. The upgraded tool is called JSI FFTBM Add-in 2007 [11] and it has been used for the sensitivity study, described in this chapter. It includes both FFTBM and FFTBM-SM. As already mentioned, the difference between FFTBM and FFTBM-SM is that in the latter the signals are symmetrized to eliminate the edge effect in calculating the average amplitude by signal mirroring (AAm).

25 frequency. The user can use the interpolated data for visual checking about the agreement of 26 the original signal and the interpolated signal. The amplitude spectra give the possibility to 27 compare the spectra between different signals. Information on the AA dependency on the 28 cut frequency is used to check if the cut frequency is selected properly. Usually the 29 dependency is not so big, therefore by default AA at the selected *f*cut frequencies is 30 calculated: minimal AA (AAmin) at frequency (when *f*cut > 0.05 *f*max 31 ) which gives the minimum AA, JSI FFTBM Add-in 2007 provides additional information on interpolated data of the signals used, the difference signals, the amplitude spectra and the AA dependency on the cut frequency. The user can use the interpolated data for visual checking about the agreement of the original signal and the interpolated signal. The amplitude spectra give the possibility to compare the spectra between different signals. Information on the AA dependency on the cut frequency is used to check if the cut frequency is selected properly. Usually the dependency is not so big, therefore by default AA at the selected *f*cut frequencies is calculated:

	- **•** average AA (AAavg) is calculated as the average AA at all cut frequencies,
	- **•** maximal AA (AAmax) at the frequency (when *f*cut > 0.05 fmax) which gives the maximum AA,
	- **•** 5 percentile AA (AA05) at frequency *f*cut = 0.05 *f*max,
	- **•** 50 percentile AA (AA50) at frequency *f*cut = 0.5 *f*max,
	- **•** 100 percentile AA (AA100) considering all amplitudes (for *f*cut = *f*max).

This gives an indication if AA is significantly dependent on the cut frequency. In principle, AA should not be much different when half or all frequencies from the amplitude spectrum are considered for the AA calculation. If this is not the case, deeper insight into AA is needed 50 percentile AA (AA50) at frequency *f*cut = 0.5 *f*max 4 ,

100 percentile AA (AA100) considering all amplitudes (for *f*cut = *f*max 5 ).

2 AA,

to judge if there is a spurious contribution present in the signal. For example, in the case of measured data, noise may be present in the signal. In the case of code calculated digital data, noise is not present in the signals, therefore the whole amplitude spectrum is recommended for the AA calculation. Nevertheless, the user should be aware that AA depends on the cut frequency and that the result may change when not all frequencies are considered. Typically higher frequency components have lower amplitudes than lower frequencies, therefore the lower frequency content is always used for the AA calculation (only higher frequency components are filtered). 6 This gives an indication if AA is significantly dependent on the cut frequency. In principle, 7 AA should not be much different when half or all frequencies from the amplitude spectrum 8 are considered for the AA calculation. If this is not the case, deeper insight into AA is 9 needed to judge if there is a spurious contribution present in the signal. For example, in the 10 case of measured data, noise may be present in the signal. In the case of code calculated 11 digital data, noise is not present in the signals, therefore the whole amplitude spectrum is 12 recommended for the AA calculation. Nevertheless, the user should be aware that AA 13 depends on the cut frequency and that the result may change when not all frequencies are 14 considered. Typically higher frequency components have lower amplitudes than lower

8 Fourier Transform

maximal AA (AAmax) at the frequency (when *f*cut > 0.05 fmax 1 ) which gives the maximum

#### **2.4. Time dependent average amplitude** 15 frequencies, therefore the lower frequency content is always used for the AA calculation 16 (only higher frequency components are filtered).

In the ref. [12] the influence of the time window selection was studied. Instead of a few phenomenological windows a series of narrow windows (phases) could be selected. This gives the possibility to get the time dependency of the average amplitude. The increasing time interval was defined as a set of time intervals each increased for the duration of one narrow time window and the last time interval being the whole transient duration time. By increasing the time interval we see how the average amplitude changes with the time progression as it is shown in Figure 2. The average amplitude was calculated by the original FFTBM not consid‐ ering the edge effect. Therefore the average amplitude shown in Figure 2(b) first increases and then partly decreases in spite of the discrepancy present all this time during the temperature increase shown in Figure 2(a). 17 **2.4. Time dependent average amplitude**  18 In the ref. [12] the influence of the time window selection was studied. Instead of a few 19 phenomenological windows a series of narrow windows (phases) could be selected. This 20 gives the possibility to get the time dependency of the average amplitude. The increasing 21 time interval was defined as a set of time intervals each increased for the duration of one 22 narrow time window and the last time interval being the whole transient duration time. By 23 increasing the time interval we see how the average amplitude changes with the time 24 progression as it is shown in Figure 2. The average amplitude was calculated by the original 25 FFTBM not considering the edge effect. Therefore the average amplitude shown in Figure 26 2(b) first increases and then partly decreases in spite of the discrepancy present all this time

The time dependent average amplitude is also indispensable for the sensitivity analysis. From such a time dependant average amplitude it can be easily seen when the largest influence occurs on the output parameter due to the sensitive parameter variation. 27 during the temperature increase shown in Figure 2(a). 28 The time dependent average amplitude is also indispensable for the sensitivity analysis. 29 From such a time dependant average amplitude it can be easily seen when the largest

30 influence occurs on the output parameter due to the sensitive parameter variation.

<sup>31</sup>**Fig.2.** Time trend of (a) primary pressure and the corresponding average amplitude **Figure 2.** Time trend of (a) primary pressure and (b) the corresponding average amplitude

## **3. Methods used for sensitivity analysis**

In BEMUSE, Phase II, single parameter sensitivity analyses have been proposed and performed by the participants to study the influence of different parameters (break area, gap conductivity, core pressure drops, time of scram etc.) upon the predicted large-break loss-of-coolant accident (LBLOCA) evolution [13]. The performed sensitivity studies were intended to be used as guidance for deriving uncertainties of relevant input parameters like for phase III of the BEMUSE programme.

The sensitivity analysis is concerned, generally, with the influence of inputs on the output and output variability. Generally, sensitivity analyses are conducted by defining the model and its independent and dependent variables, assigning probability density functions to each input parameter, generating an input matrix through an appropriate random sampling method, performing calculations, and assessing the influences and relative importance of each input/ output relationship.

In our demonstration case, the calculated data obtained in the Phase II of BEMUSE provided by the host organization have been used. The input matrix consists of the single parameter variation. The range of variations has been proposed by the host organization. For sensitivity calculations, only one value (minimal or maximal) of the input parameter was proposed when the range of the parameter variation was specified for selected sensitive parameter. For each single parameter variation the calculation was performed. The influence of the sensitive parameter variation has been estimated through the application of FFT based approaches.

We will call sensitivity (of the output parameter Y versus the i-th input parameter Xi ) a measure having the dimension of ∂*<sup>Y</sup>* ∂ *Xi* that is independent of the range of variation of the parameter Xi . Sensitivity is related to output variability.

We will call influence a measure of the effect of the variation of the parameter Xi on its full range (*ΔXi* ) having the dimension of ∂*<sup>Y</sup>* ∂ *Xi ΔXi* (same as Y) or more often dimensionless form

$$\frac{\partial Y}{\partial X\_i} \frac{\Delta Y}{\Delta X\_i}.$$

to judge if there is a spurious contribution present in the signal. For example, in the case of measured data, noise may be present in the signal. In the case of code calculated digital data, noise is not present in the signals, therefore the whole amplitude spectrum is recommended for the AA calculation. Nevertheless, the user should be aware that AA depends on the cut frequency and that the result may change when not all frequencies are considered. Typically higher frequency components have lower amplitudes than lower frequencies, therefore the lower frequency content is always used for the AA calculation (only higher frequency

6 This gives an indication if AA is significantly dependent on the cut frequency. In principle, 7 AA should not be much different when half or all frequencies from the amplitude spectrum 8 are considered for the AA calculation. If this is not the case, deeper insight into AA is 9 needed to judge if there is a spurious contribution present in the signal. For example, in the 10 case of measured data, noise may be present in the signal. In the case of code calculated 11 digital data, noise is not present in the signals, therefore the whole amplitude spectrum is 12 recommended for the AA calculation. Nevertheless, the user should be aware that AA 13 depends on the cut frequency and that the result may change when not all frequencies are 14 considered. Typically higher frequency components have lower amplitudes than lower 15 frequencies, therefore the lower frequency content is always used for the AA calculation

8 Fourier Transform

maximal AA (AAmax) at the frequency (when *f*cut > 0.05 fmax 1 ) which gives the maximum

In the ref. [12] the influence of the time window selection was studied. Instead of a few phenomenological windows a series of narrow windows (phases) could be selected. This gives the possibility to get the time dependency of the average amplitude. The increasing time interval was defined as a set of time intervals each increased for the duration of one narrow time window and the last time interval being the whole transient duration time. By increasing the time interval we see how the average amplitude changes with the time progression as it is shown in Figure 2. The average amplitude was calculated by the original FFTBM not consid‐ ering the edge effect. Therefore the average amplitude shown in Figure 2(b) first increases and then partly decreases in spite of the discrepancy present all this time during the temperature

18 In the ref. [12] the influence of the time window selection was studied. Instead of a few 19 phenomenological windows a series of narrow windows (phases) could be selected. This 20 gives the possibility to get the time dependency of the average amplitude. The increasing 21 time interval was defined as a set of time intervals each increased for the duration of one 22 narrow time window and the last time interval being the whole transient duration time. By 23 increasing the time interval we see how the average amplitude changes with the time 24 progression as it is shown in Figure 2. The average amplitude was calculated by the original 25 FFTBM not considering the edge effect. Therefore the average amplitude shown in Figure 26 2(b) first increases and then partly decreases in spite of the discrepancy present all this time

The time dependent average amplitude is also indispensable for the sensitivity analysis. From such a time dependant average amplitude it can be easily seen when the largest influence

0.0

In BEMUSE, Phase II, single parameter sensitivity analyses have been proposed and performed by the participants to study the influence of different parameters (break area, gap conductivity,

0 2000 4000 6000 8000 Time (s)

accuracy

(b) AA - cladding temperature

0.5

1.0

1.5

28 The time dependent average amplitude is also indispensable for the sensitivity analysis. 29 From such a time dependant average amplitude it can be easily seen when the largest

occurs on the output parameter due to the sensitive parameter variation.

exp RELAP5

30 influence occurs on the output parameter due to the sensitive parameter variation.

<sup>31</sup>**Fig.2.** Time trend of (a) primary pressure and the corresponding average amplitude **Figure 2.** Time trend of (a) primary pressure and (b) the corresponding average amplitude

components are filtered).

2 AA,

increase shown in Figure 2(a).

(a) Cladding temperature (K)

400

600

800

1000

1200

**2.4. Time dependent average amplitude**

17 **2.4. Time dependent average amplitude** 

16 (only higher frequency components are filtered).

27 during the temperature increase shown in Figure 2(a).

0 2000 4000 6000 8000 Time (s)

**3. Methods used for sensitivity analysis**

 5 percentile AA (AA05) at frequency *f*cut = 0.05 *f*max 3 , 50 percentile AA (AA50) at frequency *f*cut = 0.5 *f*max 4 ,

58 Fourier Transform - Signal Processing and Physical Sciences

100 percentile AA (AA100) considering all amplitudes (for *f*cut = *f*max 5 ).

In our sensitivity analysis the influences were determined. Please note that classical measures of influences are: Pearson's or Spearman's Correlation Coefficients, Standardised Regression Coefficients, etc. [14]. In FFT based approaches the AA is a dimensionless number, showing influences in terms of the average amplitude obtained in the frequency domain which represents the physical influence (e.g. temperature or pressure change).

## **3.1. Test description**

The LOFT L2-5 test was selected for this demonstration because a huge amount of data was available [15]. The reference and sensitivity calculations of the LOFT L2-5 test were performed in the phase II of the BEMUSE research program. The nuclear LOFT integral test facility is a scale model of a pressurized water reactor. The objective of the test was to simulate a loss of coolant accident (LOCA) caused by a double-ended, off-shear guillotine cold leg rupture coupled with a loss of off-site power. The experiment was initiated by opening the quick opening blowdown valves in the broken loop hot and cold legs. The reactor scrammed and emergency core cooling systems started their injection. After initial heatup the core was quenched at 65 s, following the core reflood. The low pressure injection system (LPIS) was stopped at 107.1 s, after the experiment was considered complete. In total 14 calculations from 13 organizations were performed. For more detailed information on the calculations the reader is referred to [4, 15].

#### **3.2. Sensitivity calculations description**

The series of sensitivity calculations with assigned parameters was proposed to participants. For each parameter the host organization recommended the value to be used. The short description of the cases to be analysed is given in Table 1.


**Table 1.** List of sensitivity analyses and proposed parameter variations (adapted per Table 6 in ref. [15])


In Table 2 sensitivity calculations performed by 14 participants are shown. Each row presents sensitivity calculations S-1 to S-15. If the calculation is performed with recommended values, the sign √ is used. If another value has been used, the value of the sensitive parameter is indicated. If the sensitivity calculation has not been performed, the cell is shaded grey.

\* no information given in Ref. [15], but high certainty that recommended values were used;

√ - recommended value used

quenched at 65 s, following the core reflood. The low pressure injection system (LPIS) was stopped at 107.1 s, after the experiment was considered complete. In total 14 calculations from 13 organizations were performed. For more detailed information on the calculations the reader

The series of sensitivity calculations with assigned parameters was proposed to participants. For each parameter the host organization recommended the value to be used. The short

is referred to [4, 15].

**ID Parameter**

S-6 Core Pressure Drop

(UP)

S-7

**3.2. Sensitivity calculations description**

60 Fourier Transform - Signal Processing and Physical Sciences

S-4 Presence of Crud 0.15 mm

CCFL at Upper Tie Plate (UTP) and/or connection upper plenum

S-9 Time of Scram RC + 1 s

RC: value used in Reference Case

description of the cases to be analysed is given in Table 1.

**Recommended**

S-2 Gap Conductivity RC x 0.2 Only in the hot rod in the hot channel (zone 4). S-3 Gap Thickness RC x 3 Only in the hot rod in the hot channel (zone 4).

S-5 Fuel Conductivity RC x 0.4 Only in the hot rod in the hot channel (zone 4).

DPtot=(DPtot)RC x 1.3

S-10 Maximum Linear Power RC x 1.5 Only in the hot rod in the hot channel (zone 4).

**Table 1.** List of sensitivity analyses and proposed parameter variations (adapted per Table 6 in ref. [15])

RC + D:

Range not assigned

S-14 HPIS Failure Failure of HPIS.

S-15 LPIS injection initiated RC + 30 s Delay in starting LPIS injection.

**values (RV) Description**

S-1 Break Area RC x 1.15 Tube diameter from reactor pressure vessel to break point shall

e.g. Al2O3.

S-8 Decay Power RC x 1.25 The decay power has to be 25% bigger than in the reference case.

S-11 Accumulator Pressure RC - 0.5 MPa Set point of accumulator pressure 0.5 MPa lower than the set

S-12 Accumulator Liquid Mass RC x 0.7 Accumulator liquid mass shall be 0.7 times the value in the

S-13 Pressurizer Level RC - 0.5 m Pressurizer level shall be 0.5 m lower than in the reference case.

power.

reference case.

be modified in respect to RC.

Consideration of 0.15 mm of crud in hot rod in hot channel with thermal conductivity that is characteristic of ceramic material,

The pressure drop across the core shall increase (decrease) of an amount D to obtain a total pressure drop that is 30% bigger than

The power curve shall follow the imposed trend that implies full power till RC + 1 sec and after that shall followed the decay

the total pressure drop of the reference case.

point in the base case (= 4.29 MPa).

Counter courant flow limitation (CCFL) is nodalization dependent. Each participant can propose a solution.

**Table 2.** Sensitivity calculations performed by participants

#### **3.3. Figures of merit for sensitivity analysis**

Same figures of merit were proposed for the original FFTBM and improved FFTBM-SM. In the case of signal mirroring the additional index m is used to distinguish the FFTBM-SM from FFTBM. The first figure of merit is the average amplitude (AA or AAm), which tells how the single input parameter variation (or combination of input parameter variations) influences the output parameter. As there are several sensitive input parameters, several participants performing sensitivity runs and more selected output parameters, three additional figures of merit were proposed. The average amplitude of the participant sensitivity runs (AAp or AAmp) is used to judge which sensitivity runs set is more influential to the input parameter variations. AAp or AAmp is calculated as the average of AAs for the participant sensitivity runs (15 in our specific case). The average amplitude of the sensitivity runs for the same sensitive parameter (AAs or AAms) is used to judge how influential (in average) the sensitive parameter is in calculations performed by different participants. AAs or AAms is calculated as the average of AAs of participants for the same sensitivity run (14 in our specific case). Finally, the total average amplitude (AAt or AAmt) is the average AA or average AAm obtained from all sensitivity runs performed by all participants.

## **4. Application of FFT based approches to sensitivity analysis**

In this section the application of the original FFTBM and FFTBM-SM is demonstrated. As was explained in Subsection 2.1, two digital signals (reference and sensitive) of the same duration are used for calculating the average amplitude, which is a measure of the influence of the sensitive parameter variation. An example of calculating AAm is given first. Then the single value influence (based on the average amplitude) for the whole time window is given. Finally, the time dependent influence is presented for the sensitivity runs. We conclude this section with the discussion.

## **4.1. AAm figure of merit example**

In the example the influence of sensitive parameters on three output parameters for the interval 0-119.5 s (whole transient time) is shown for calculations of participant P-14. The participant P-14 provided for each time trend 241 samples. This means that the sampling frequency was 2 Hz. The input values for FFTBM-SM were therefore the following: ffix=2 Hz, fcut=2 Hz, tb=0 s and te=119.5 s (time 119.5 s was selected because some users provided the last data point at a value slightly smaller than 120 s). Per sampling theorem (Eq. (1)) at least 478 data points are needed. However, the minimum number of points per FFTBM-SM is 512. Table 3 shows that requesting more samples than required has a minor influence on the results.

To get some qualitative impression on the influence of input parameters on output parameters judged by FFTBM-SM, Figure 3 shows time trends of P-14 upper plenum pressure, primary side mass and cladding temperature. Visually it may be indicated that the upper plenum pressure is the most influenced by the break flow (S-1). On the other hand, hot rod parameters (S-2, S-3, S-4 and S-5) and accumulator initial mass (S-12) do not have significant influence on upper plenum pressure (see Figures 3(a2), 3(a3), 3(a4) and 3(a5)). The primary side mass inventory is the most influenced by the gap conductivity (S-2), fuel conductivity (S-5), and the accumulator liquid mass (S-12). When comparing S-2 and S-5 calculations, one may indicate that the S-5 calculation is a bit closer to the reference calculation. On the other hand, comparing the S-2 and S-12 calculation, in the case of S-2 the difference is absolutely larger than S-12,


**Table 3.** Average amplitude AAm for the whole time interval for participant P-14 calculated by FFFTB-SM as a function of the number of samples N

however in case of S-12 the difference is present a longer time than in the case of S-2. The calculated AAm is comparable.

To see this in more detail, Figure 4 shows part of the magnitude difference signal spectra <sup>|</sup>*<sup>Δ</sup>*˜*<sup>F</sup>* ( *<sup>f</sup> <sup>n</sup>*)| for S-2 and S-12, which are used for the calculation of AAdif per Eq. (5). Please note that fmax is 2.14 Hz. However, summing of amplitudes up to 0.2 Hz contributes more than 90% to total AAdif (for S-2 the sum is 11.26 out of 12.43 and for S-12 the sum is 10.83 out of 11.79). Summing amplitudes up to 0.05 Hz (representing 13 samples out of 513) contributes more than 80% to total AAdif (for S-2 the sum is 10.16 out of 12.43 and for S-12 the sum is 9.57 out of 11.79). One may see that the zero frequency component (mean value of difference signal in the time domain) is larger for S-2 than S-12 and that due to this contribution finally S-2 is judged as more influential than S-12.

## **4.2. Single value influence for whole time window**

output parameter. As there are several sensitive input parameters, several participants performing sensitivity runs and more selected output parameters, three additional figures of merit were proposed. The average amplitude of the participant sensitivity runs (AAp or AAmp) is used to judge which sensitivity runs set is more influential to the input parameter variations. AAp or AAmp is calculated as the average of AAs for the participant sensitivity runs (15 in our specific case). The average amplitude of the sensitivity runs for the same sensitive parameter (AAs or AAms) is used to judge how influential (in average) the sensitive parameter is in calculations performed by different participants. AAs or AAms is calculated as the average of AAs of participants for the same sensitivity run (14 in our specific case). Finally, the total

**4. Application of FFT based approches to sensitivity analysis**

requesting more samples than required has a minor influence on the results.

In this section the application of the original FFTBM and FFTBM-SM is demonstrated. As was explained in Subsection 2.1, two digital signals (reference and sensitive) of the same duration are used for calculating the average amplitude, which is a measure of the influence of the sensitive parameter variation. An example of calculating AAm is given first. Then the single value influence (based on the average amplitude) for the whole time window is given. Finally, the time dependent influence is presented for the sensitivity runs. We conclude this section

In the example the influence of sensitive parameters on three output parameters for the interval 0-119.5 s (whole transient time) is shown for calculations of participant P-14. The participant P-14 provided for each time trend 241 samples. This means that the sampling frequency was 2 Hz. The input values for FFTBM-SM were therefore the following: ffix=2 Hz, fcut=2 Hz, tb=0 s and te=119.5 s (time 119.5 s was selected because some users provided the last data point at a value slightly smaller than 120 s). Per sampling theorem (Eq. (1)) at least 478 data points are needed. However, the minimum number of points per FFTBM-SM is 512. Table 3 shows that

To get some qualitative impression on the influence of input parameters on output parameters judged by FFTBM-SM, Figure 3 shows time trends of P-14 upper plenum pressure, primary side mass and cladding temperature. Visually it may be indicated that the upper plenum pressure is the most influenced by the break flow (S-1). On the other hand, hot rod parameters (S-2, S-3, S-4 and S-5) and accumulator initial mass (S-12) do not have significant influence on upper plenum pressure (see Figures 3(a2), 3(a3), 3(a4) and 3(a5)). The primary side mass inventory is the most influenced by the gap conductivity (S-2), fuel conductivity (S-5), and the accumulator liquid mass (S-12). When comparing S-2 and S-5 calculations, one may indicate that the S-5 calculation is a bit closer to the reference calculation. On the other hand, comparing the S-2 and S-12 calculation, in the case of S-2 the difference is absolutely larger than S-12,

or AAmt) is the average AA or average AAm obtained from all

average amplitude (AAt

with the discussion.

**4.1. AAm figure of merit example**

sensitivity runs performed by all participants.

62 Fourier Transform - Signal Processing and Physical Sciences

In the example presented in Section 4.1, AAm for the P-14 calculation was determined. In this section all fourteen calculations are considered, and both FFTBM and FFTBM-SM are used for calculating all figures of merit presented in Section 3.3. The obtained results for the sensitive parameter influence on the three output parameters (upper plenum pressure, primary side mass inventory and rod surface temperature) are shown in Tables 4 through 6.

The qualitative comparison between FFTBM and FFTBM-SM results showed that the agree‐ ment is quite good. This is expected as at the end of the transient the influence of the sensitive parameter is generally insignificant, resulting that in the difference signal the edge is very

14 Fourier Transform

1 When looking calculations, the most influenced upper plenum calculation is P-3. To judge 2 how the calculation is influenced by the sensitive parameters variation, the average 3 amplitude of the participant sensitivity runs (AAp or AAmp) is used. Besides P-3 the P-5 4 calculation was also judged as the much influenced by the sensitive parameter variations. 5 Both methods qualitatively give the same results for the average amplitude of the 6 participant sensitivity runs. The total average amplitude (AAt or AAmt) show the overall

8 plenum pressure. The higher the value is the higher is the influence. The ratio between AAmt

9 and AAt to be 1.88 also tells what the contribution of the edge effect is in average.

11 **Figure 3.** surface temperature for S-1, S-2, S-3, S-5 and S-12 with AA Participant P-14 time trends of (a) primary pressure, (b) primary side mass and (c) rod surface temperature <sup>m</sup> for S-1, S-2, S-3, S-5 and S-12 with AAm

10 Fig.3. Participant P-14 time trends of (a) primary pressure, (b) primary side mass and (c) rod

small or not present at all. The edge is still present in the reference signal, but because it is used for the normalization it has no impact on the ranking of parameters and so the qualitative agreement between FFTBM and FFTBM-SM is good. This is not the case for the quantitative agreement as the normalization directly impacts the average amplitude. The average ampli‐ tudes obtained by both FFTBM and FFTBM-SM suggest that the most influential parameter for the upper plenum pressure is in all calculations the break flow area (S-1). To judge how influential the parameter is, the average amplitude of the sensitivity runs for the same sensitive parameter (AAs or AAms) obtained both by FFTBM and FFTBM-SM show that the variations of the break flow area (S-1), pressurizer level (S-13), core pressure drop (S-6) and presence of crud (S-4) the most influence the output parameter upper plenum pressure. The only difference between the FFTBM and FFTBM-SM results is that ranks for S-6 and S-4 are changed.


**Table 4.**  Influence of sensitive parameters on upper plenum pressure in time interval 0-119.5 s as judged by (a) original FFTBM and (b) improved FFTBM by signal mirroring

small or not present at all. The edge is still present in the reference signal, but because it is used for the normalization it has no impact on the ranking of parameters and so the qualitative agreement between FFTBM and FFTBM-SM is good. This is not the case for the quantitative agreement as the normalization directly impacts the average amplitude. The average ampli‐ tudes obtained by both FFTBM and FFTBM-SM suggest that the most influential parameter for the upper plenum pressure is in all calculations the break flow area (S-1). To judge how influential the parameter is, the average amplitude of the sensitivity runs for the same sensitive

10 Fig.3. Participant P-14 time trends of (a) primary pressure, (b) primary side mass and (c) rod 11 **Figure 3.** surface temperature for S-1, S-2, S-3, S-5 and S-12 with AA Participant P-14 time trends of (a) primary pressure, (b) primary side mass and (c) rod surface temperature <sup>m</sup>

14 Fourier Transform

0 40 80 120 Time (s)

(b2) Mass (%) - AAm=0.12

0 40 80 120 Time (s)

(b3) Mass (%) - AAm=0.06

0 40 80 120 Time (s)

(b4) Mass (%) - AAm=0.11

0 40 80 120 Time (s)

> ref sen (S-12)

(b5) Mass (%) - AAm=0.12

0 40 80 120 Time (s)

ref sen (S-1)

ref sen (S-2)

ref sen (S-3)

ref sen (S-5)

0 40 80 120 Time (s)

(c2) Temperature (K) - AAm=0.80

0 40 80 120 Time (s)

(c3) Temperature (K) - AAm=0.49

0 40 80 120 Time (s)

(c4) Temperature (K) - AAm=0.94

0 40 80 120 Time (s)

(c5) Temperature (K) - AAm=0.48 ref sen (S-12)

0 40 80 120 Time (s)

ref sen (S-5)

ref sen (S-2)

ref sen (S-3)

(c1) Temperature (K) - AAm=0.33

ref sen (S-1)

Table 4

(b1) Mass (%) - AAm=0.10

1 When looking calculations, the most influenced upper plenum calculation is P-3. To judge 2 how the calculation is influenced by the sensitive parameters variation, the average 3 amplitude of the participant sensitivity runs (AAp or AAmp) is used. Besides P-3 the P-5 4 calculation was also judged as the much influenced by the sensitive parameter variations. 5 Both methods qualitatively give the same results for the average amplitude of the 6 participant sensitivity runs. The total average amplitude (AAt or AAmt) show the overall 7 influence of the sensitive parameters variations of all calculations on the output upper 8 plenum pressure. The higher the value is the higher is the influence. The ratio between AAmt

9 and AAt to be 1.88 also tells what the contribution of the edge effect is in average.

0 40 80 120 Time (s)

(a2) Pressure (MPa) - AAm=0.03

0 40 80 120 Time (s)

(a3) Pressure (MPa) - AAm=0.02

0 40 80 120 Time (s)

(a4) Pressure (MPa) - AAm=0.03

0 40 80 120 Time (s)

(a5) Pressure (MPa) - AAm=0.03

0 40 80 120 Time (s)

for S-1, S-2, S-3, S-5 and S-12 with AAm

(a1) Pressure (MPa) - AAm=0.09

ref sen (S-1)

64 Fourier Transform - Signal Processing and Physical Sciences

ref sen (S-2)

ref sen (S-3)

ref sen (S-5)

ref sen (S-12) When looking calculations, the most influenced upper plenum calculation is P-3. To judge how the calculation is influenced by the sensitive parameters variation, the average amplitude of the participant sensitivity runs (AAp or AAmp) is used. Besides P-3 the P-5 calculation was also judged as the much influenced by the sensitive parameter variations. Both methods qualita‐ tively give the same results for the average amplitude of the participant sensitivity runs. The total average amplitude (AAt or AAmt) show the overall influence of the sensitive parameters variations of all calculations on the output upper plenum pressure. The higher the value is the higher is the influence. The ratio between AAmt and AAt to be 1.88 also tells what the contri‐ bution of the edge effect is in average.

**Figure 4.** Magnitude difference signal spectra for S-2 and S-12 runs of P-14 participants

The average amplitudes shown in Table 5, obtained by both FFTBM and FFTBM-SM suggest that the most influential parameter for the primary side mass inventory when considering all calculations is the accumulator liquid mass (S-12). Both the AAs and AAms indicate the accumulator liquid mass (S-12), break flow area (S-1), accumulator pressure (S-11) and pressurizer level (S-13) the most influential sensitive parameters on the primary side mass inventory.

When looking the calculations, the most influenced primary side mass inventory calculation is P-3. The AAp and AAmp indicate as the second most influenced the P-13 calculation. Again both methods qualitatively give very similar results for the average amplitude of the partici‐ pant sensitivity runs. The AAt and AAmt show that the overall influence of the sensitive parameters variations of all calculations on the output primary side mass inventory is higher than on the upper plenum pressure. The ratio between AAmt and AAt is 1.55, indicating that the primary side mass inventory is less influenced by the edge effect (see also Figure3).

The average amplitudes shown in Table 6, obtained by both FFTBM and FFTBM-SM suggest that the most influential parameter for the rod surface temperature when considering all calculations is the fuel conductivity (S-5). Both AAs and AAms indicate the fuel conductivity (S-5) and gap conductivity (S-2) as the most influential. Significant influences have also the gap thickness (S-3), maximum linear power (S-10), break flow area (S-1) and the accumulator liquid mass (S-12) and a few others. The only difference between the FFTBM and FFTBM-SM results is that the ranks for S-1 and S-12 are changed.


When looking calculations, the most influenced upper plenum calculation is P-3. To judge how the calculation is influenced by the sensitive parameters variation, the average amplitude of the participant sensitivity runs (AAp or AAmp) is used. Besides P-3 the P-5 calculation was also judged as the much influenced by the sensitive parameter variations. Both methods qualita‐ tively give the same results for the average amplitude of the participant sensitivity runs. The

variations of all calculations on the output upper plenum pressure. The higher the value is the higher is the influence. The ratio between AAmt and AAt to be 1.88 also tells what the contri‐

0.00 0.05 0.10 0.15 0.20

Frequency (Hz)

The average amplitudes shown in Table 5, obtained by both FFTBM and FFTBM-SM suggest that the most influential parameter for the primary side mass inventory when considering all calculations is the accumulator liquid mass (S-12). Both the AAs and AAms indicate the accumulator liquid mass (S-12), break flow area (S-1), accumulator pressure (S-11) and pressurizer level (S-13) the most influential sensitive parameters on the primary side mass

When looking the calculations, the most influenced primary side mass inventory calculation is P-3. The AAp and AAmp indicate as the second most influenced the P-13 calculation. Again both methods qualitatively give very similar results for the average amplitude of the partici‐

parameters variations of all calculations on the output primary side mass inventory is higher than on the upper plenum pressure. The ratio between AAmt and AAt is 1.55, indicating that the primary side mass inventory is less influenced by the edge effect (see also Figure3).

The average amplitudes shown in Table 6, obtained by both FFTBM and FFTBM-SM suggest that the most influential parameter for the rod surface temperature when considering all calculations is the fuel conductivity (S-5). Both AAs and AAms indicate the fuel conductivity (S-5) and gap conductivity (S-2) as the most influential. Significant influences have also the gap thickness (S-3), maximum linear power (S-10), break flow area (S-1) and the accumulator liquid mass (S-12) and a few others. The only difference between the FFTBM and FFTBM-SM

**Figure 4.** Magnitude difference signal spectra for S-2 and S-12 runs of P-14 participants

or AAmt) show the overall influence of the sensitive parameters

and AAmt show that the overall influence of the sensitive

S-2 S-12 Table 5

total average amplitude (AAt

inventory.

pant sensitivity runs. The AAt

results is that the ranks for S-1 and S-12 are changed.

Amplitude

bution of the edge effect is in average.

66 Fourier Transform - Signal Processing and Physical Sciences

**Table 5.**  Influence of sensitive parameters on primary side mass inventory in time interval 0-119.5 s as judged by (a) original FFTBM and (b) improved FFTBM by signal mirroring

When looking the calculations, the most influenced rod surface rod temperature calculation is P-10. The AAp and AAmp indicate that the next two most influenced are the P-2 and P-3 calculation. Again both methods qualitatively give pretty similar results for the average amplitude of the participant sensitivity runs. The AAt and AAmt show that the overall influence of the sensitive parameters variations of all calculations on the output rod surface temperature Table 6


**Table 6.** Influence of sensitive parameters on rod surface temperature in time interval 0-119.5 s as judged by (a) original FFTBM and (b) improved FFTBM by signal mirroring

is higher than on the primary side mass inventory. The ratio between AAmt and AAt is 1.17, indicating that the rod surface temperature is the least influenced by the edge effect (see also Figure 3).

## **4.3. Time dependent influence**

is higher than on the primary side mass inventory. The ratio between AAmt and AAt

**Table 6.** Influence of sensitive parameters on rod surface temperature in time interval 0-119.5 s as judged by (a)

Figure 3).

Table 6

(a) Application of FFTBM - rod surface temperature, time interval 0 - 119.5 s

68 Fourier Transform - Signal Processing and Physical Sciences

0.25-0.45 >0.45

Legend: 0.3-0.5 >0.5

original FFTBM and (b) improved FFTBM by signal mirroring

(b) Application of FFTBM-SM - rod surface temperature, time interval 0 - 119.5 s

ID-S S-1 S-2 S-3 S-4 S-5 S-6 S-7 S-8 S-9 S-10 S-11 S-12 S-13 S-14 S-15 ID-P AA AA AA AA AA AA AA AA AA AA AA AA AA AA AA AAp P-1 0.323 0.498 0.424 0.085 0.318 0.189 0.122 0.194 0.253 0.413 0.283 0.342 0.204 0.104 0.129 0.259 P-2 0.323 0.403 0.608 0.493 0.312 0.301 0.282 0.495 0.220 1.679 0.097 0.099 0.174 0.422 P-3 0.432 0.451 0.476 0.406 0.703 0.496 0.415 0.386 0.266 0.426 0.198 0.212 0.247 0.184 0.378 P-4 0.368 0.252 0.445 0.185 0.210 0.289 0.527 0.195 0.388 0.207 0.328 0.107 0.292 P-5 0.184 0.284 0.174 0.330 0.308 0.328 0.268 P-6 0.622 0.379 0.402 0.167 0.543 0.415 0.444 0.349 0.157 0.483 0.392 0.210 0.360 0.172 0.166 0.351 P-7 0.340 0.207 0.239 0.358 0.261 0.281 P-8 0.446 0.396 0.305 0.150 0.351 0.227 0.283 0.329 0.337 0.360 0.117 0.434 0.149 0.171 0.133 0.279 P-9 0.229 0.407 0.343 0.135 0.386 0.208 0.089 0.152 0.255 0.207 0.208 0.212 0.160 0.000 0.072 0.204 P-10 0.345 0.848 0.555 0.762 0.264 0.349 0.339 0.683 0.311 0.238 0.401 0.080 0.307 0.422 P-11 0.207 0.507 0.639 0.335 0.243 0.358 0.681 0.204 0.437 0.224 0.239 0.279 0.104 0.142 0.328 P-12 0.344 0.598 0.419 0.216 0.543 0.192 0.110 0.252 0.339 0.825 0.232 0.187 0.245 0.145 0.211 0.324 P-13 0.253 0.535 0.339 0.124 0.464 0.343 0.238 0.197 0.349 0.221 0.290 0.256 0.111 0.090 0.160 0.265 P-14 0.257 0.668 0.420 0.267 0.738 0.206 0.205 0.113 0.173 0.386 0.197 0.380 0.236 0.204 0.114 0.304 AAs 0.334 0.498 0.432 0.209 0.499 0.270 0.213 0.294 0.282 0.428 0.258 0.397 0.222 0.145 0.158AAt=0.314

ID-S S-1 S-2 S-3 S-4 S-5 S-6 S-7 S-8 S-9 S-10 S-11 S-12 S-13 S-14 S-15 ID-P AAm AAm AAm AAm AAm AAm AAm AAm AAm AAm AAm AAm AAm AAm AAm AAmp P-1 0.396 0.605 0.499 0.103 0.383 0.229 0.149 0.231 0.302 0.490 0.338 0.415 0.247 0.127 0.155 0.311 P-2 0.389 0.475 0.717 0.605 0.368 0.358 0.348 0.542 0.265 0.913 0.121 0.120 0.214 0.418 P-3 0.508 0.515 0.516 0.481 0.861 0.580 0.486 0.461 0.305 0.489 0.234 0.250 0.285 0.217 0.442 P-4 0.438 0.285 0.504 0.226 0.255 0.350 0.557 0.230 0.459 0.249 0.402 0.131 0.340 P-5 0.227 0.360 0.213 0.402 0.385 0.400 0.331 P-6 0.736 0.467 0.444 0.200 0.656 0.481 0.526 0.412 0.183 0.525 0.452 0.247 0.423 0.201 0.196 0.410 P-7 0.411 0.251 0.288 0.416 0.316 0.336 P-8 0.528 0.456 0.372 0.188 0.432 0.274 0.342 0.404 0.414 0.449 0.149 0.483 0.189 0.209 0.165 0.337 P-9 0.293 0.492 0.418 0.172 0.506 0.264 0.114 0.194 0.311 0.238 0.264 0.270 0.202 0.000 0.091 0.255 P-10 0.410 0.982 0.625 0.930 0.307 0.402 0.391 0.768 0.367 0.284 0.472 0.094 0.353 0.491 P-11 0.244 0.579 0.727 0.412 0.270 0.442 0.829 0.243 0.523 0.278 0.286 0.342 0.126 0.170 0.391 P-12 0.414 0.680 0.487 0.260 0.667 0.231 0.128 0.309 0.383 0.637 0.282 0.227 0.293 0.170 0.251 0.361 P-13 0.302 0.604 0.401 0.148 0.563 0.411 0.291 0.238 0.412 0.266 0.351 0.292 0.134 0.109 0.177 0.313 P-14 0.330 0.799 0.488 0.295 0.937 0.256 0.258 0.143 0.205 0.449 0.239 0.480 0.302 0.255 0.147 0.372 AAms 0.402 0.585 0.498 0.251 0.609 0.324 0.258 0.353 0.338 0.469 0.309 0.382 0.269 0.175 0.189 AAmt=0.366

indicating that the rod surface temperature is the least influenced by the edge effect (see also

is 1.17,

The results presented in Section 4.2 give information on the accumulated influence of sensitive parameters for the whole transient duration (single value figures of merit). Additional insight into the results is obtained from the time dependent average amplitudes for each single variation of parameters. They provide information how the influence changes during the transient progression.

Figure 5 shows the comparison between the FFTBM and FFTBM-SM results for the P-14 sensitive calculations shown in Figure 3. It is shown how the sensitive parameter influence changes during the transient progression. The judged quantitative influence in Figure 5 reflects well what is seen during the visual observation of Figure 3, in which 5 out of 15 sensitive parameter variations for the three output parameters for the P-14 calculation are shown. Please note that the FFT based approaches are especially to be used when there are several calculations (fourteen in our case) with several sensitive parameter variations (fifteen in our case) to judge the influence of the sensitive parameters in an uniform way.

When looking the output parameter upper plenum pressure, both FFTBM (excluding period when edge effect significantly contributes to average amplitude) and FFTBM-SM clearly show when during the transient the parameter was influential. For all parameters shown in Figure 5 the major influence was during the first 30 seconds when the pressure was dropping. In the case of the S-1 parameter the influence was the largest (see Figure 5(a1)), but still not extremely significant. For parameters S-2, S-3, S-5 and S-12 the total influence is small. The values of average amplitudes up to 0.03 are small. This is confirmed by Figures 3(a2), 3(a3), 3 (a4) and 3 (a5) which show that the reference and sensitive signals for the upper plenum pressure practically match each other.

When looking the output parameter primary side mass inventory, the influence of the sensitive parameters is also quite small. Parameter S-1 is the most influential in the beginning of the transient (see also Figure 3(b1)), while all other shown sensitive parameters (S-2, S-3, S-5 and S-12) become more influential later into the transient. This is in agreement with the Figures 3(b2), 3(b3), 3(b4) and 3(b5), in which the differences in the first 20 seconds are practically not visible.

Finally, when looking the output parameter rod surface temperature, the influence of the sensitive parameters is the largest among the selected output parameters as shown by the plots and the average amplitude trends. The variation of the break flow area (S-1) having the largest influence on the upper plenum pressure has a lower influence on the rod surface temperature than the sensitive parameters S-2, S-3, S-5 and S-12. This is logically as the break area size directly impacts the upper plenum pressure.

From Figure 5(c1) it can be seen that the influence of S-1 on the rod surface temperature is judged to be in the beginning of the transient and at around 60 s. When comparing the sensitive signal to the reference signal in Figure 3(c1), in the beginning for the sensitive signal a slower temperature increase with under predicted peak and earlier temperature decrease (rod quench) at around 60 s can be seen. At other times the trends are similar. In the case of S-2 the temperature is over predicted and the quench is delayed. Therefore besides the initial jump Use of Fast Fourier Transform for Sensitivity Analysis

19

1 **Fig.5.** Participant P-14 influence of S-1, S-2, S-3, S-5, S-11 and S-12 sensitive parameter 2 variations on (a) primary pressure and (b) primary side mass and (c) cladding temperature **Figure 5.** Participant P-14 influence of S-1, S-2, S-3, S-5, S-11 and S-12 sensitive parameter variations on (a) primary pressure and (b) primary side mass and (c) cladding temperature

3 From Figure 5(c1) it can be seen that the influence of S-1 on the rod surface temperature is 4 judged to be in the beginning of the transient and at around 60 s. When comparing the

the AAm is still increasing till 20 s and for the time duration of the rod quench delay. For parameter S-3 it can be seen that its influence is between the S-1 and the S-2 influence, what can be confirmed from Figures 3(c1), 3(c2) and 3(c3). The influence of S-5 on the rod surface 5 sensitive signal to the reference signal in Figure 3(c1), in the beginning for the sensitive a 6 slower temperature increase with under predicted peak and earlier temperature decrease 7 (rod quench) at around 60 s can be seen. At other times the trends are similar. In the case of 8 S-2 the temperature is over predicted and the quench is delayed. Therefore besides the

temperature is the largest what can be easily confirmed when comparing Figure 3(c4) and Figure 5(c4). The influence of S-11 is smaller than S-1 because S-11 influences only the time of rod quenching. Finally, S-12 shows that the larger discrepancy in the times when the rod quench starts causes also larger values of average amplitudes. Also when comparing the average amplitudes obtained by FFTBM and FFTBM-SM it can be seen that they agree pretty well for the output parameters primary side mass inventory and the rod surface temperature, because the edge present in the periodic signal is relatively smaller from the edge present in the upper plenum pressure periodic signal. 20 Fourier Transform 1 initial jump the AAm is still increasing till 20s and for the time duration of the rod quench 2 delay. For parameter S-3 it can be seen that its influence is between the S-1 and the S-2 3 influence, what can be confirmed from Figures 3(c1), 3(c2) and 3(c3). The influence of S-5 on 4 the rod surface temperature is the largest what can be easily confirmed when comparing 5 Figure 3(c4) and Figure 5(c4). The influence of S-11 is smaller than S-1 because S-11 6 influences only the time of rod quenching. Finally, S-12 shows that the larger discrepancy in 7 the times when the rod quench starts causes also larger values of average amplitudes.Also

Figure 6(a) shows the comparison of the rod surface temperature reference calculations to experimental data. One may see that the calculations differ and that they do not exactly match the experimental data. Therefore the reader should always keep in mind that the direct comparison of sensitive signals (see Figure 6(b)) obtained by different participants could not answer in which calculation the sensitive parameter is the most influential. 8 when comparing the average amplitudes obtained by FFTBM and FFTBM-SM it can be seen 9 that they agree pretty well for the output parameters primary side mass inventory and the 10 rod surface temperature, because the edge present in the periodic signal is relatively smaller 11 from the edge present in the upper plenum pressure periodic signal. 12 Figure 6(a) shows the comparison of the rod surface temperature reference calculations to 13 experimental data. One may see that the calculations differ and that they do not exactly

14 match the experimental data. Therefore the reader should always keep in mind that the 15 direct comparison of sensitive signals (see Figure6(b)) obtained by different participants

16 could not answer in which calculation the sensitive parameter is the most influential.

19 However, by calculating the average amplitude the sensitive runs by different participants 20 could be compared as it is shown in the Figures 7, 8 and 9 for the output parameters upper **Figure 6.** Comparison of (a) reference calculations with experimental value and (b) sensitive runs for S-2 sensitive pa‐ rameter variation

17 **Fig.6.** Comparison of (a) reference calculations with experimental value nd (b) sensitive runs

18 for S-2 sensitive parameter variation

21 plenum pressure, primary side mass inventory and rod surface temperature, respectively. 22 For each of the output parameters the most influential parameters are shown as identified 23 from Tables 4, 5 and 6. Figure 7 for the upper plenum pressure shows that for the majority 24 of calculations the influences of the same sensitive parameter variation are similar. There are 25 only a few calculations significantly deviating, for example P-4 for S-1 variation, P-3 and P-26 10 for S-13 variation, P-3 for S-6 variation and P-11 for S-4 variation. Figure 8 for the primary 27 side mass inventory shows that the P3 calculation was more sensitive to variations than 28 other calculations. In the case of the S-4 sensitive parameter variation the P-11 calculation 29 significantly deviates from other calculations and the reason may be that the code model is 30 used outside its validation range [13]. However, by calculating the average amplitude the sensitive runs by different participants could be compared as it is shown in the Figures 7, 8 and 9 for the output parameters upper plenum pressure, primary side mass inventory and rod surface temperature, respectively. For each of the output parameters the most influential parameters are shown as identified from Tables 4, 5 and 6. Figure 7 for the upper plenum pressure shows that for the majority of calculations the influences of the same sensitive parameter variation are similar. There are only a few calculations significantly deviating, for example P-4 for S-1 variation, P-3 and P-10 for S-13 variation, P-13 for S-6 variation and P-11 for S-4 variation. Figure 8 for the primary side mass inventory shows that the P3 calculation was more sensitive to variations than other

the AAm is still increasing till 20 s and for the time duration of the rod quench delay. For parameter S-3 it can be seen that its influence is between the S-1 and the S-2 influence, what can be confirmed from Figures 3(c1), 3(c2) and 3(c3). The influence of S-5 on the rod surface

Use of Fast Fourier Transform for Sensitivity Analysis

0.0

0.2

0.0

0.2

0.0

0.2

0.0

0.2

0.0

0.1

0.1

0.1

0.1

0 40 80 120

AA AAm

(b2) S-2 influence on PS mass inventory

0 40 80 120

(b3) S-3 influence on PS mass inventory

AA AAm

AA AAm

AA AAm

0 40 80 120

(b4) S-5 influence on PS mass inventory

0 40 80 120

(b5) S-12 influence on PS mass inventory

0 40 80 120

1 **Fig.5.** Participant P-14 influence of S-1, S-2, S-3, S-5, S-11 and S-12 sensitive parameter 2 variations on (a) primary pressure and (b) primary side mass and (c) cladding temperature

**Figure 5.** Participant P-14 influence of S-1, S-2, S-3, S-5, S-11 and S-12 sensitive parameter variations on (a) primary

3 From Figure 5(c1) it can be seen that the influence of S-1 on the rod surface temperature is 4 judged to be in the beginning of the transient and at around 60 s. When comparing the 5 sensitive signal to the reference signal in Figure 3(c1), in the beginning for the sensitive a 6 slower temperature increase with under predicted peak and earlier temperature decrease 7 (rod quench) at around 60 s can be seen. At other times the trends are similar. In the case of 8 S-2 the temperature is over predicted and the quench is delayed. Therefore besides the

(b1) S-1 influence on PS mass inventory

AA AAm 0.0 0.2 0.4 0.6 0.8 1.0

0.0 0.2 0.4 0.6 0.8 1.0

0.0 0.2 0.4 0.6 0.8 1.0

0.0 0.2 0.4 0.6 0.8 1.0

0.0 0.2 0.4 0.6 0.8 1.0 0 40 80 120

(c2) S-2 influence on RS temperature

0 40 80 120

(c3) S-3 influence on RS temperature

0 40 80 120

(c4) S-5 influence on RS temperature

0 40 80 120

(c5) S-12 influence on RS temperature

0 40 80 120

(c1) S-1 influence on RS temperature

AA AAm

AA AAm

> AA AAm

AA AAm

AA AAm

0.1

0.2

AA AAm

70 Fourier Transform - Signal Processing and Physical Sciences

AA AAm

AA AAm

AA AAm

AA AAm

pressure and (b) primary side mass and (c) cladding temperature

0.00

0.00

0.10

0.00

0.10

0.00

0.10

0.00

0.05

0.05

0.05

0.05

0.10

0 40 80 120

(a2) S-2 influence on UP pressure

0 40 80 120

(a3) S-3 influence on UP pressure

0 40 80 120

(a4) S-5 influence on UP pressure

0 40 80 120

(a5) S-12 influence on UP pressure

0 40 80 120

(a1) S-1 influence on UP pressure

0.05

0.10

19

Use of Fast Fourier Transform for Sensitivity Analysis

calculations. In the case of the S-4 sensitive parameter variation the P-11 calculation signifi‐ cantly deviates from other calculations and the reason may be that the code model is used outside its validation range [13].

Figure 9 shows that some parameters are more influential in the beginning of the transient (e.g. S-5), some in the middle of the transient (e.g. S-12) and that in the last part normally there is no significant influence (exception is the calculation P-2 for the S-12 sensitive parameter in which the rod surface temperature did not quench due to no accumulator injection when it was supposed to inject).

21

2 6 and (d) S-4 sensitive parameter variations 3 Figure 8 shows that some parameters are more influential in the beginning of the transient **Figure 7.** Upper plenum pressure time dependent AAm of participants for (a) S-1, (b) S-13, (c) S-6 and (d) S-4 sensitive parameter variations

1 **Fig.7.** Upper plenum pressure time dependent AAm of participants for (a) S-1, (b) S-13, (c) S-

4 (e.g. S-5), some in the middle of the transient (e.g. S-12) and that in the last part normally 5 there is no significant influence (exception is the calculation P-2 for the S-12 sensitive 6 parameter in which the rod surface temperature did not quench due to no accumulator

7 injection when it was supposed to inject).

22 Fourier Transform

calculations. In the case of the S-4 sensitive parameter variation the P-11 calculation signifi‐ cantly deviates from other calculations and the reason may be that the code model is used

Figure 9 shows that some parameters are more influential in the beginning of the transient (e.g. S-5), some in the middle of the transient (e.g. S-12) and that in the last part normally there is no significant influence (exception is the calculation P-2 for the S-12 sensitive parameter in which the rod surface temperature did not quench due to no accumulator injection when it

0

0.00

(d)

**Figure 7.** Upper plenum pressure time dependent AAm of participants for (a) S-1, (b) S-13, (c) S-6 and (d) S-4 sensitive

1 **Fig.7.** Upper plenum pressure time dependent AAm of participants for (a) S-1, (b) S-13, (c) S-

3 Figure 8 shows that some parameters are more influential in the beginning of the transient 4 (e.g. S-5), some in the middle of the transient (e.g. S-12) and that in the last part normally 5 there is no significant influence (exception is the calculation P-2 for the S-12 sensitive 6 parameter in which the rod surface temperature did not quench due to no accumulator

0.05

0.10

0.15

AAm (-) 0.20

0.25

0 40 80 120

P-1 P-3 P-6 P-8 P-9 P-11 P-12 P-13 P-14

S-4

P-1 P-2 P-3 P-4 P-6 P-8 P-9 P-10 P-11 P-12 P-13 P-14

Time (s)

0 40 80 120

Time (s)

0.05

0.1

0.15

AAm (-)

(b)

0.2

S-13

0.25

21

outside its validation range [13].

72 Fourier Transform - Signal Processing and Physical Sciences

P-13 P-14

Use of Fast Fourier Transform for Sensitivity Analysis

P-1 P-2 P-3 P-4 P-5 P-6 P-7 P-8 P-9 P-10 P-11 P-12

0 40 80 120

P-1 P-2 P-3 P-4 P-5 P-6 P-7 P-8 P-9 P-10 P-11 P-12

S-6

0 40 80 120

Time (s)

2 6 and (d) S-4 sensitive parameter variations

7 injection when it was supposed to inject).

Time (s)

P-13 P-14

was supposed to inject).

S-1

0.00

0.00

parameter variations

(c)

0.05

0.10

0.15

AAm (-) 0.20

0.25

(a)

0.05

0.10

0.15

AAm (-) 0.20

0.25

1 **Fig.8.** Primary side mass inventory time dependent AAm of participants for (a) S-12, (b) S-1, 2 (c) S-11, (d) S-4, (e) S-13 and (f) S-5 sensitive parameter variations **Figure 8.** Primary side mass inventory time dependent AAm of participants for (a) S-12, (b) S-1, (c) S-11, (d) S-4, (e) S-13 and (f) S-5 sensitive parameter variations

Use of Fast Fourier Transform for Sensitivity Analysis

23

2 **Figure 9.** 3, (d) S-10, (e) S-1 and (f) S-12 sensitive parameter variations Rod surface temperature time dependent AAm of participants for (a) S-5, (b) S-2, (c) S-3, (d) S-10, (e) S-1 and (f) S-12 sensitive parameter variations

1 **Fig.9.** Rod surface temperature time dependent AAm of participants for (a) S-5, (b) S-2, (c) S-

## **4.4. Discussion**

23

Use of Fast Fourier Transform for Sensitivity Analysis

74 Fourier Transform - Signal Processing and Physical Sciences

P-1 P-2 P-3 P-4 P-6 P-8 P-9 P-10 P-11 P-12 P-13 P-14

0 40 80 120

P-1 P-2 P-3 P-4 P-6 P-8 P-9 P-10 P-11 P-12 P-13 P-14

0 40 80 120

P-1 P-2 P-3 P-4 P-5 P-6 P-7 P-8 P-9 P-10 P-11 P-12

0 40 80 120

Time (s)

Time (s)

P-13 P-14

0

0

0

(f)

1 **Fig.9.** Rod surface temperature time dependent AAm of participants for (a) S-5, (b) S-2, (c) S-2 **Figure 9.** 3, (d) S-10, (e) S-1 and (f) S-12 sensitive parameter variations Rod surface temperature time dependent AAm of participants for (a) S-5, (b) S-2, (c) S-3, (d) S-10, (e) S-1 and

0.2

0.4

0.6

AAm (-)

0.8

1

S-12

0.2

0.4

0.6

AAm (-)

(d)

0.8

1

P-14

S-10

0 40 80 120

P-1 P-2 P-3 P-4 P-5 P-6 P-7 P-8 P-9 P-11 P-12 P-13

0 40 80 120

P-1 P-2 P-3 P-4 P-6 P-8 P-9 P-10 P-11 P-12 P-13 P-14

Time (s)

0 40 80 120

Time (s)

P-1 P-2 P-3 P-5 P-6 P-8 P-9 P-10 P-11 P-12 P-13 P-14

Time (s)

0.2

0.4

0.6

AAm (-)

(b)

0.8

1

S-2

Time (s)

0

0

0

0.2

0.4

0.6

AAm (-)

(e)

0.8

1

S-1

(f) S-12 sensitive parameter variations

(c)

0.2

0.4

0.6

AAm (-) 0.8

1

S-3

0.2

0.4

0.6

AAm (-)

(a)

0.8

1

S-5

In the application for each participant each sensitivity run in the set was compared to his reference calculation and the average amplitude, which is the figure of merit for FFT based approaches, was used to judge how each output parameter is sensitive to the selected input parameter. For brevity reasons, only some examples were given in Figures 6 through 9. Nevertheless, these examples are sufficient to demonstrate that the results for the whole transient interval, shown in Tables 4, 5 and 6 may not be always sufficient for judging the sensitivity. For example, for the output parameter rod surface temperature there is higher interest in the value of the maximum rod surface temperature rather the average influence on the rod surface temperature, therefore it is important to know how the influence changes with time. Also, some parameters are more influential at the beginning and the others later in the calculation.

The results suggest that the FFT based approaches are especially appropriate for a quick sensitivity analysis in which several calculations need to be compared. It is very appropriate also due to the inherent feature, which integrates the contribution of the parameter variation with progressing transient time.

In addition, the average amplitude of participant sensitivity runs for the participants (AAp or AAmp) and the average amplitude for the same sensitive parameter (AAs or AAms) were calculated. These measures could be used for ranking purposes. In this way information on the most influential input parameter and which participant calculation is the most sensitive to variations is obtained. Finally, different output parameters could be compared between each other regarding the influence of input parameters of all participants (AAt or AAmt). Quantita‐ tively it is judged that the most influenced output parameter is the rod surface temperature and the least influenced the upper plenum pressure.

## **5. Conclusions**

The study using FFTBM-SM and FFTBM was performed to show that the FFT based ap‐ proaches could be used for sensitivity analyses. The LOFT L2-5 test, which simulates the large break loss of coolant accident, was used in the frame of the BEMUSE programme for sensitivity runs. In total 15 sensitivity runs were performed by 14 participants.

It can be concluded that with FFTBM-SM the analyst can get a good picture of the influence of the single parameter variation to the results throughout the transient. Some sensitive parameters are more influential at the beginning and the others later in the calculation. Due to the edge effect FFTBM-SM is advantageous for time dependent sensitivity calculations with respect to FFTBM, while for the whole transient duration (average sensitivity during whole transient) in general also FFTBM gives consistent results. FFT based approaches could be also used to quantify the influence of several parameter variations on the results. However, the influential parameters could not be identified nor the direction of the influence.

The results suggest that the FFT based approaches are especially appropriate for a quick assessment of a sensitivity analysis in which several calculations need to be compared or the influence of single sensitive parameters needs to be ranked. Such a sensitivity analysis could provide information which are the most influential parameters and how influential the input parameters are on the selected output parameters and when they influence during a transient.

## **Author details**

Andrej Prošek and Matjaž Leskovar

Reactor Engineering Division, Jožef Stefan Institute, Ljubljana, Slovenia

## **References**


[8] Kunz RF, Kasmala GF, Mahaffy JH, Murray CJ. On the automated assessment of nu‐ clear reactor systems code accuracy. Nucl Eng Des. 2002; 211(2-3):245-272. PubMed PMID: WOS:000174067600010. English.

The results suggest that the FFT based approaches are especially appropriate for a quick assessment of a sensitivity analysis in which several calculations need to be compared or the influence of single sensitive parameters needs to be ranked. Such a sensitivity analysis could provide information which are the most influential parameters and how influential the input parameters are on the selected output parameters and when they influence during a transient.

[1] Perez M, Reventos F, Batet L, Guba A, Toth I, Mieusset T, et al. Uncertainty and sen‐ sitivity analysis of a LBLOCA in a PWR Nuclear Power Plant: Results of the Phase V of the BEMUSE programme. Nucl Eng Des. 2011;241(10):4206-4222. PubMed PMID:

[2] Prošek A, Leskovar M. Application of fast Fourier transform for accuracy evaluation of thermal-hydraulic code calculation. In: S. NG, editor. Fourier transforms - ap‐

[3] de Crecy A, Bazin P, Glaeser H, Skorek T, Joucla J, Probst P, et al. Uncertainty and sensitivity analysis of the LOFT L2-5 test: Results of the BEMUSE programme. Nucl

[4] Prošek A, Leskovar M, Mavko B. Quantitative assessment with improved fast Fouri‐ er transform based method by signal mirroring. Nucl Eng Des. 2008;238(10):

[5] Prošek A, D'Auria F, Mavko B. Review of quantitative accuracy assessments with fast Fourier transform based method (FFTBM). Nucl Eng Des. 2002;217(1-2):179-206.

[6] Smith SW. The Scientist and Engineer's Guide to Digital Signal Processing. San Die‐

[7] Lapendes DN. Dictionary of scientific and technical terms. Second Edition ed. New York, St. Louis, San Francisco, Auckland, Bogotá, Düsseldorf, Johannesburg, London, Madrid, Mexico, Montreal, New Delhi, Panama, Paris, Sãn Paulo, Singapore, Sydney,

go, California: California Technical Publishing, Second Edition; 1999.

Tokyo, Toronto, : McGraw-Hill Book Company; 1978.

Reactor Engineering Division, Jožef Stefan Institute, Ljubljana, Slovenia

proach to scientific principles. Rijeka: In-Tech; 2011. p. 447-68.

**Author details**

**References**

Andrej Prošek and Matjaž Leskovar

76 Fourier Transform - Signal Processing and Physical Sciences

WOS:000295956100015. English.

Eng Des. 2008;238(12):3561-3578.

2668-2677.


## **Henstock-Kurzweil Integral Transforms and the Riemann-Lebesgue Lemma Henstock-Kurzweil Integral Transforms and the Riemann-Lebesgue Lemma**

Francisco J. Mendoza-Torres, Ma. Guadalupe Morales-Macías, Salvador Sánchez-Perales and Juan Alberto Escamilla-Reyna Francisco J. Mendoza-Torres1, Ma. Guadalupe Morales-Macías2, Salvador Sánchez-Perales3, and Juan Alberto Escamilla-Reyna1

Additional information is available at the end of the chapter

Additional information is available at the end of the chapter

**1. Introduction**

Let *f* be a function defined on a closed interval [*a*, *b*] in the extended real line **R**, its Fourier transform at *s* ∈ **R** is defined as

$$
\hat{f}(\mathbf{s}) = \int\_{a}^{b} e^{-i\mathbf{x}\mathbf{s}} f(\mathbf{x}) d\mathbf{x}.\tag{1}
$$

The classical Riemann-Lebesgue Lemma states that

$$\lim\_{|s|\to\infty} \int\_{a}^{b} e^{-i\mathbf{x}\mathbf{s}} f(\mathbf{x}) d\mathbf{x} = \mathbf{0},\tag{2}$$

whenever *<sup>f</sup>* <sup>∈</sup> *<sup>L</sup>*1([*a*, *<sup>b</sup>*]).

We consider important to study analogous results about this lemma due to the following reasons:


$$\int\_{a}^{b} h(\mathbf{x}s)f(\mathbf{x})d\mathbf{x}.\tag{3}$$

• In some cases the expression (1) exists and the expression (2) is true even if the function *f* is not Lebesgue integrable.

Thus, a variant of the Riemann-Lebesgue Lemma is to get conditions for the functions *f* and *h* which ensure that (3) is well defined and satisfies

$$\lim\_{||s|\to\infty} \int\_{a}^{b} h(\infty) f(\infty) d\mathfrak{x} = 0. \tag{4}$$

Some results of this type and related results are found in [1], [2], [3], and [4].

In the space of Henstock-Kurzweil integrable functions over **R**, *HK*(**R**), the Fourier transform does not always exist. In [5] was proven that *e*−*i*(·)*<sup>s</sup> f* is Henstock-Kurzweil integrable under certain conditions and that, in general, does not satisfy the Riemann-Lebesgue Lemma. Subsequently, it was shown in [6], [3] and [4] that the Fourier transform exists and the equation (2) is true when −∞ = *a*, *b* = ∞ and *f* belongs to *BV*0(**R**), the space of bounded variation functions that vanish at infinity. A special case arises when *f* is in the intersection of functions of bounded variation and Henstock-Kurzweil integrable functions.

There exist Henstock-Kurzweil (HK) integrable functions *<sup>f</sup>* which *<sup>f</sup>* <sup>∈</sup> *HK*(*I*) \ *<sup>L</sup>*1(*I*) such that (2) is not fulfilled, when *I* is a bounded interval. In [7], Zygmund exhibited Henstock-Kurzweil integrable functions such that their Fourier coefficients do not tend to zero. In [8] are given necessary and sufficient conditions in order to *<sup>b</sup> <sup>a</sup> f*(*x*)*gn*(*x*)*dx* −→ *b <sup>a</sup> f*(*x*)*g*(*x*)*dx*, for all *f* ∈ *HK*([*a*, *b*]). Thus, we will prove that the Fourier transform has the asymptotic behavior:

$$\hat{f}(s) = o(s), \text{ as } |s| \to \infty,$$

where *<sup>f</sup>* <sup>∈</sup> *HK*(*I*) \ *<sup>L</sup>*1(*I*)

Moreover in [9], Titchmarsh proved that it is the best possible approximation for functions with improper Riemann integral.

This chapter is divided into 5 sections; we present the main results we have obtained in recent years: [3], [4], [6] and [10]. In this section we introduce basic concepts and important theorems about the Henstock-Kurzweil integral and bounded variation functions. In the second part of this study we prove some generalizations about the convergence of integrals of products in the completion of the space *HK*([*a*, *b*]), *HK*([*a*, *b*]), where [*a*, *b*] can be a bounded or unbounded interval. As a consequence, some results related to the Riemann-Lebesgue Lemma in the context of the Henstock-Kurzweil integral are proved over bounded intervals. Besides, for elements in the completion of the space of Henstock-Kurzweil integrable functions, we get a similar result to the Riemann-Lebesgue property for the Dirichlet Kernel, as well as the asymptotic behavior of the *n*-th partial sum of Fourier series. In the third section, we consider a complex function *g* defined on certain subset of **R**2. Many functions on functional analysis are integrals of the form Γ(*s*) = <sup>∞</sup> <sup>−</sup><sup>∞</sup> *<sup>f</sup>*(*t*)*g*(*t*,*s*)*dt*. We study the function Γ when *f* belongs to *BV*0(**R**) and *g*(*t*, ·) is continuous for all *t*. The integral we use is Henstock-Kurzweil integral. There are well known results about existence, continuity and differentiability of Γ, considering the Lebesgue theory. In the HK integral context there are results about this too, for example, Theorems 12.12 and 12.13 from [11]. But they all need the stronger condition that the function *f*(*t*)*g*(*t*,*s*) is bounded by a HK integrable function. We give more conditions for existence, continuity and differentiability of Γ. Finally we give some applications such as some properties about the convolution of the Fourier and Laplace transforms.

In section 4, we exhibit a family of functions in *HK*(**R**) included in *BV*0(**R**) \ *<sup>L</sup>*1(**R**). At the last section we get a version of Riemann-Lebesgue Lemma for bounded variation functions that vanish at infinity. With this result we get properties for the Fourier transform of functions in *BV*0(**R**): it is well defined, is continuous on **R** − {0}, and vanishes at ±∞, as classical results. Moreover, we obtain a result on pointwise inversion of the Fourier transform.

#### **1.1. Basic concepts and nomenclature**

We will refer to a finite or infinite interval if its Lebesgue measure is finite or infinite. Let *I* ⊂ **R** be a closed interval, finite or infinite. A *partition P* of *I* is a increasing finite collection of points {*t*0, *t*1, ..., *tn*} ⊂ *I* such that if *I* is a compact interval [*a*, *b*], then *t*<sup>0</sup> = *a* and *tn* = *b*; if *I* = [*a*, ∞), *t*<sup>0</sup> = *a*; and if *I* = (−∞, *b*] then *tn* = *b*.

Let us consider *I* ⊂ **R** as a closed interval finite. A *tagged partition* of *I* is a set of ordered pairs {[*ti*−1, *ti*],*si*}*<sup>n</sup> <sup>i</sup>*=<sup>1</sup> where it is assigned a point *si* ∈ [*ti*−1, *ti*], which is called a *tag* of [*ti*−1, *ti*]. With this concept we define the Henstock-Kurzweil integral on finite intervals in **R**.

**Definition 1.** *The function f* : [*a*, *b*] → **R** *is* Henstock-Kurzweil integrable *if there exists H* ∈ **R** *which satisfies the following: for each ε >* 0 *exists a function γε* : [*a*, *b*] → (0, ∞) *such that if <sup>P</sup>* <sup>=</sup> {( [*ti*−1, *ti*],*si*)}*<sup>n</sup> <sup>i</sup>*=<sup>1</sup> *is a tagged partition such that*

$$\left[\mathbf{t}\_{i-1}, \mathbf{t}\_{i}\right] \subset \left[\mathbf{s}\_{i} - \gamma\_{\varepsilon}(\mathbf{s}\_{i}), \mathbf{s}\_{i} + \gamma\_{\varepsilon}(\mathbf{s}\_{i})\right] \quad \text{for } \mathbf{i} = 1, 2, \dots, n. \tag{5}$$

*then*

2 ime knjige

*b*

asymptotic behavior:

where *<sup>f</sup>* <sup>∈</sup> *HK*(*I*) \ *<sup>L</sup>*1(*I*)

with improper Riemann integral.

*f* is not Lebesgue integrable.

*h* which ensure that (3) is well defined and satisfies

 *b a*

lim |*s*|→∞  *b a*

Some results of this type and related results are found in [1], [2], [3], and [4].

of functions of bounded variation and Henstock-Kurzweil integrable functions.

zero. In [8] are given necessary and sufficient conditions in order to *<sup>b</sup>*

ˆ

• In some cases the expression (1) exists and the expression (2) is true even if the function

Thus, a variant of the Riemann-Lebesgue Lemma is to get conditions for the functions *f* and

In the space of Henstock-Kurzweil integrable functions over **R**, *HK*(**R**), the Fourier transform does not always exist. In [5] was proven that *e*−*i*(·)*<sup>s</sup> f* is Henstock-Kurzweil integrable under certain conditions and that, in general, does not satisfy the Riemann-Lebesgue Lemma. Subsequently, it was shown in [6], [3] and [4] that the Fourier transform exists and the equation (2) is true when −∞ = *a*, *b* = ∞ and *f* belongs to *BV*0(**R**), the space of bounded variation functions that vanish at infinity. A special case arises when *f* is in the intersection

There exist Henstock-Kurzweil (HK) integrable functions *<sup>f</sup>* which *<sup>f</sup>* <sup>∈</sup> *HK*(*I*) \ *<sup>L</sup>*1(*I*) such that (2) is not fulfilled, when *I* is a bounded interval. In [7], Zygmund exhibited Henstock-Kurzweil integrable functions such that their Fourier coefficients do not tend to

*<sup>a</sup> f*(*x*)*g*(*x*)*dx*, for all *f* ∈ *HK*([*a*, *b*]). Thus, we will prove that the Fourier transform has the

*f*(*s*) = *o*(*s*), as |*s*| → ∞,

Moreover in [9], Titchmarsh proved that it is the best possible approximation for functions

This chapter is divided into 5 sections; we present the main results we have obtained in recent years: [3], [4], [6] and [10]. In this section we introduce basic concepts and important theorems about the Henstock-Kurzweil integral and bounded variation functions. In the second part of this study we prove some generalizations about the convergence of integrals of products in the completion of the space *HK*([*a*, *b*]), *HK*([*a*, *b*]), where [*a*, *b*] can be a bounded or unbounded interval. As a consequence, some results related to the Riemann-Lebesgue Lemma in the context of the Henstock-Kurzweil integral are proved over bounded intervals. Besides, for elements in the completion of the space of Henstock-Kurzweil integrable functions, we get a similar result to the Riemann-Lebesgue property for the Dirichlet Kernel, as well as the asymptotic behavior of the *n*-th partial sum of Fourier series.

*h*(*xs*)*f*(*x*)*dx*. (3)

*h*(*xs*)*f*(*x*)*dx* = 0. (4)

*<sup>a</sup> f*(*x*)*gn*(*x*)*dx* −→

$$|\Sigma\_{i=1}^{n}f(\mathbf{s}\_{i})(t\_{i}-t\_{i-1})-H|<\varepsilon.$$

*H is the integral of f over* [*a*, *b*] *and it is denoted as*

$$H = \int\_{a}^{b} f = \int\_{a}^{b} fdt$$

*It is said that a tagged partition is called γε*−*fine if satisfies* (5)*.*

This definition can be extended on infinite intervals as follows.

**Definition 2.** *Let γ* : [*a*, ∞] → (0, ∞) *be a function, we will say that the tagged partition P* = {( [*ti*−1, *ti*],*si*)}*n*+<sup>1</sup> *<sup>i</sup>*=<sup>1</sup> is *γ*−fine *if:*

*(a) t*<sup>0</sup> = *a*, *tn*+<sup>1</sup> = ∞*. (b)* [*ti*−1, *ti*] ⊂ [*si* − *γε*(*si*),*si* + *γε*(*si*) ] *for i* = 1, 2, ..., *<sup>n</sup>*. *(c)* [*tn*, ∞] ⊂ [1/*γ*(∞), ∞]*.*

Put *f*(∞) = 0 and *f*(−∞) = 0. This allows us define the integral of *f* over infinite intervals.

**Definition 3.** *It is said that the function f* : [*a*, ∞] → **R** *is Henstock-Kurzweil integrable if it satisfies the Definition 1, but the partition P must be γε*−*fine according to Definition 2.*

For functions defined on [−∞, *a*] or [−∞, +∞] the integral is defined analogously. We will denote the vector space of Henstock-Kurzweil integrable functions on *I* as *HK*(*I*)

The space of Henstock-Kurzweil integrable functions on the interval *I* = [*a*, *b*], finite or infinite interval, is a semi-normed space with the Alexiewicz semi-norm

$$||f||\_A = \sup\_{a \le x \le b} \left| \int\_a^x f(t)dt \right|. \tag{6}$$

We denote the space of functions in *HK*([*c*, *d*]) for each finite interval [*c*, *d*] in *I* as *HKloc*(*I*).

**Definition 4.** *A function f* : *I* → **R** *is a* bounded variation *function over I (finite interval) if exists a M >* 0 *such that*

$$\text{Var}(f, I) = \sup \left\{ \sum\_{i=1}^{n} |f(t\_i) - f(t\_{i-1})| : P \text{ is a partition of } I \right\} < M.$$

*Its total variation over I is* Var(*f* , *I*)*. In case I is not finite, for example* [*a*, ∞]*, it is said that f* : [*a*, ∞] → **R** *is a bounded variation function over I if there exists N >* 0 *such that*

$$\text{Var}(f, [a, t]) \le N\_\prime$$

*for all t* ≥ *a. The total variation of f on I is equal to*

$$\text{Var}(f, [a, \infty)) = \sup \left\{ \text{Var}(f, [a, t] \, ) : a \le t \right\}.\tag{7}$$

*For I* = (−∞, *b*] *the considerations are analogous.*

The set of bounded variation functions over [*a*, *b*] is denoted as *BV*([*a*, *b*]) and we will denote the space of functions *f* such that *f* ∈ *BV*([*c*, *d*]) for each compact interval [*c*, *d*] in **R** as *BVloc*(**R**). We will refer to *BV*0(**R**) as the subspace of functions *f* belong to *BV*(**R**) such that vanishing at ±∞.

At the Lemma 25 we prove that: *HK*(**R**) ∩ *BV*(**R**) ⊂ *BV*0(**R**). It is not hard prove that *BV*0(**R**) *L*1(**R**) and *BV*0(**R**) *HK*(**R**). Furthermore, there are functions in *HK*(**R**) or *<sup>L</sup>*1(**R**) but they are not in *BV*0(**R**). For example, the function *<sup>f</sup>*(*t*) defined by 0 for *<sup>t</sup>* <sup>∈</sup> (−∞, 1) and 1/*<sup>t</sup>* for *<sup>t</sup>* <sup>∈</sup> [1, <sup>∞</sup>) belongs to *BV*0(**R**) but does not belong to *<sup>L</sup>*1(**R**), neither in *HK*(**R**). In addition, other examples are the characteristic function of **Q** on a compact interval and *g*(*t*) = *t* <sup>2</sup> sin(exp(*t* <sup>2</sup>)) are in *HK*(**R**) \ *BV*0(**R**).

We consider the completion of *HK*([*a*, *b*]) as

4 ime knjige

{( [*ti*−1, *ti*],*si*)}*n*+<sup>1</sup>

*(a) t*<sup>0</sup> = *a*, *tn*+<sup>1</sup> = ∞*.*

*(c)* [*tn*, ∞] ⊂ [1/*γ*(∞), ∞]*.*

*exists a M >* 0 *such that*

vanishing at ±∞.

Var(*f* , *I*) = sup

*for all t* ≥ *a. The total variation of f on I is equal to*

*For I* = (−∞, *b*] *the considerations are analogous.*

*<sup>i</sup>*=<sup>1</sup> is *γ*−fine *if:*

*(b)* [*ti*−1, *ti*] ⊂ [*si* − *γε*(*si*),*si* + *γε*(*si*) ] *for i* = 1, 2, ..., *<sup>n</sup>*.

**Definition 2.** *Let γ* : [*a*, ∞] → (0, ∞) *be a function, we will say that the tagged partition P* =

Put *f*(∞) = 0 and *f*(−∞) = 0. This allows us define the integral of *f* over infinite intervals. **Definition 3.** *It is said that the function f* : [*a*, ∞] → **R** *is Henstock-Kurzweil integrable if it satisfies*

For functions defined on [−∞, *a*] or [−∞, +∞] the integral is defined analogously. We will

The space of Henstock-Kurzweil integrable functions on the interval *I* = [*a*, *b*], finite or

 *x a*

*f*(*t*)*dt* 


Var(*f* , [*a*, ∞)) = sup {Var(*f* , [*a*, *t*] ) : *a* ≤ *t*} . (7)

. (6)

*< M*.

*a*≤*x*≤*b*

We denote the space of functions in *HK*([*c*, *d*]) for each finite interval [*c*, *d*] in *I* as *HKloc*(*I*). **Definition 4.** *A function f* : *I* → **R** *is a* bounded variation *function over I (finite interval) if*

*Its total variation over I is* Var(*f* , *I*)*. In case I is not finite, for example* [*a*, ∞]*, it is said that*

Var(*f* , [*a*, *t*]) ≤ *N*,

The set of bounded variation functions over [*a*, *b*] is denoted as *BV*([*a*, *b*]) and we will denote the space of functions *f* such that *f* ∈ *BV*([*c*, *d*]) for each compact interval [*c*, *d*] in **R** as *BVloc*(**R**). We will refer to *BV*0(**R**) as the subspace of functions *f* belong to *BV*(**R**) such that

denote the vector space of Henstock-Kurzweil integrable functions on *I* as *HK*(*I*)


*f* : [*a*, ∞] → **R** *is a bounded variation function over I if there exists N >* 0 *such that*

*the Definition 1, but the partition P must be γε*−*fine according to Definition 2.*

infinite interval, is a semi-normed space with the Alexiewicz semi-norm

 *n* ∑ *i*=1

$$\{ [\{ f\_k \}] : \{ f\_k \} \text{ is a Cauchy sequence in } HK([a, b]) \},$$

where the convergence is respect Alexiewicz norm, and will be denoted by *HK*([*a*, *b*]). It is possible to prove that *HK*([*a*, *b*]) is isometrically isomorphic to the subspace of distributions each of which is the distributional derivative of a continuous function, see [12]. The indefinite integral of *<sup>f</sup>* = [{ *fk*}] <sup>∈</sup> *HK*([*a*, *<sup>b</sup>*]) is defined as

$$\int\_{a}^{\chi} f = \lim\_{k \to \infty} \int\_{a}^{\chi} f\_{k}.$$

Thus, *HK*([*a*, *b*]) is a Banach space with the Alexiewicz norm (6). The completion is also defined in [13]. Besides, basic results of the integral continue being true on the completion. More details see [12].

To facilitate reading, we recall the following results. The first one is a well known result, and it can be found for example in [14] and [15].

**Theorem 5.** *Let f be a real function defined on* **N** × **N***. If* lim*n*→*n*<sup>0</sup> *f*(*k*, *n*) = *ψ*(*k*) *exists for each <sup>k</sup>*, *and* lim*k*→*k*<sup>0</sup> *<sup>f</sup>*(*k*, *<sup>n</sup>*) = *<sup>ϕ</sup>*(*n*) *converges uniformly on n, then*

$$\lim\_{k \to k\_0} \lim\_{n \to n\_0} f(k, n) = \lim\_{n \to n\_0} \lim\_{k \to k\_0} f(k, n).$$

**Theorem 6.** *[16, Theorem 33.1] Suppose X is a normed space, Y is a Banach space and that* {*Tn*} *is a sequence of bounded linear operators from X into Y*. *Then the conditions: i*) {||*Tn*||} *is bounded and ii*) {*Tn*(*x*)} *is convergent for each x* ∈ *Z*, *where Z is a dense subset on X implies that for each x* ∈ *X*, *the sequence* (*Tn*(*x*)) *is convergent in Y and the linear operator T* : *X* → *Y defined by T*(*x*) = lim*n*→<sup>∞</sup> *Tn*(*x*) *is bounded.*

**Theorem 7.** *[17] If g is a HK integrable function on* [*a*, *b*] ⊆ **R** *and f is a bounded variation function on* [*a*, *b*]*, then f g is HK integrable on* [*a*, *b*] *and*

$$\left| \int\_{a}^{b} fg \right| \leq \inf\_{t \in [a,b]} |f(t)| \left| \int\_{a}^{b} g(t)dt \right| + ||g||\_{[a,b]} \text{Var}(f, [a,b]).$$

**Theorem 8.** *[11, Hake's Theorem] ϕ* ∈ *HK*([*a*, ∞]) *if and only if for each b, ε such that b > a <sup>b</sup>* <sup>−</sup> *<sup>a</sup> <sup>&</sup>gt; <sup>ε</sup> <sup>&</sup>gt;* 0, *it follows that <sup>ϕ</sup>* <sup>∈</sup> *HK*([*<sup>a</sup>* <sup>+</sup> *<sup>ε</sup>*, *<sup>b</sup>*]) *and* lim *<sup>ε</sup>*→0, *<sup>b</sup>*→<sup>∞</sup> *b <sup>a</sup>*+*<sup>ε</sup> ϕ*(*t*)*dt exists. In this case, this limit is* <sup>∞</sup> *<sup>a</sup> ϕ*(*t*)*dt.*

**Theorem 9.** *[11, Chartier-Dirichlet's Test] Let f and g be functions defined on* [*a*, ∞)*. Suppose that*


*Then f g* ∈ *HK*([*a*, ∞))*.*

Moreover, by Multiplier Theorem, Hake's Theorem and Chartier-Dirichlet Test, we have the following lemma.

**Lemma 10.** *Let f* , *g* : [*a*, ∞] → **R***. Suppose that f* ∈ *BV*0([*a*, ∞]), *ϕ* ∈ *HK*([*a*, *b*]) *for every b > a*, *and* Φ(*t*) = *<sup>t</sup> <sup>a</sup> ϕdu is bounded on* [*a*, ∞). *Then ϕ f* ∈ *HK*([*a*, *b*]),

$$\int\_{a}^{\infty} \varrho f dt = -\int\_{a}^{\infty} \Phi(t) df(t)$$

*and*

$$\left| \int\_{a}^{\infty} \varrho f dt \right| \leq \sup\_{a$$

*Similar results are valid for the cases* [−∞, ∞] *and* [−∞, *a*].

Let *I* = [*a*, *b*] and *E* ⊂ *I*. We say that the function *F* : *I* → **R** is in *ACδ*(*E*) if for each *>* 0 there exist *ηε <sup>&</sup>gt;* 0 and a gauge *δε* on *<sup>E</sup>* such that if {(*xi*, *yi*)}*<sup>N</sup> <sup>i</sup>* is a (*δε*, *E*)−fine subpartition of *E* such that ∑*<sup>N</sup> <sup>i</sup>* (*yi* <sup>−</sup> *xi*) *<sup>&</sup>lt; ηε*, then <sup>∑</sup>*<sup>N</sup>* <sup>1</sup> |*F*(*xi*) − *F*(*yi*)| *<* . On the other hand, *F* belongs to the class *ACGδ*(*I*) if there exists a sequence {*En*}<sup>∞</sup> <sup>1</sup> of sets in *<sup>I</sup>* such that *<sup>I</sup>* <sup>=</sup> <sup>∪</sup><sup>∞</sup> *<sup>n</sup>*=1*En* and *F* ∈ *ACδ*(*En*) for each *n* ∈ **N**. A characterization of this type of functions is the following.

**Theorem 11.** *A function f* ∈ *HK*(*I*) *if and only if there exists a function F* ∈ *ACG<sup>δ</sup> such that F* = *f a.e.*

**Theorem 12.** *[18, Theorem 4] Let a*, *b* ∈ **R***. If h* : **R** × [*a*, *b*] → **C** *is such that*


*Then H* := <sup>∞</sup> <sup>−</sup><sup>∞</sup> *<sup>h</sup>*(*t*, ·)*dt belongs to ACG<sup>δ</sup> on* [*a*, *<sup>b</sup>*] *and H* (*s*) = <sup>∞</sup> <sup>−</sup><sup>∞</sup> *<sup>D</sup>*2*h*(*t*,*s*)*dt for almost all s* ∈ (*a*, *b*)*, iff,*

$$\int\_{s}^{t} \int\_{-\infty}^{\infty} \mathcal{D}\_{2}h(t,s)dtds = \int\_{-\infty}^{\infty} \int\_{s}^{t} \mathcal{D}\_{2}h(t,s)dsdt$$

*for all* [*s*, *t*] ⊆ [*a*, *b*]*. In particular,*

6 ime knjige

*limit is* <sup>∞</sup>

*<sup>a</sup> ϕ*(*t*)*dt.*

*Then f g* ∈ *HK*([*a*, ∞))*.*

following lemma.

of *E* such that ∑*<sup>N</sup>*

= *f a.e.*

*Then H* := <sup>∞</sup>

*s* ∈ (*a*, *b*)*, iff,*

*and* Φ(*t*) = *<sup>t</sup>*

*and*

*F* 

**Theorem 8.** *[11, Hake's Theorem] ϕ* ∈ *HK*([*a*, ∞]) *if and only if for each b, ε such that b > a*

**Theorem 9.** *[11, Chartier-Dirichlet's Test] Let f and g be functions defined on* [*a*, ∞)*. Suppose that*

Moreover, by Multiplier Theorem, Hake's Theorem and Chartier-Dirichlet Test, we have the

**Lemma 10.** *Let f* , *g* : [*a*, ∞] → **R***. Suppose that f* ∈ *BV*0([*a*, ∞]), *ϕ* ∈ *HK*([*a*, *b*]) *for every b > a*,

Let *I* = [*a*, *b*] and *E* ⊂ *I*. We say that the function *F* : *I* → **R** is in *ACδ*(*E*) if for each *>* 0

*F* ∈ *ACδ*(*En*) for each *n* ∈ **N**. A characterization of this type of functions is the following. **Theorem 11.** *A function f* ∈ *HK*(*I*) *if and only if there exists a function F* ∈ *ACG<sup>δ</sup> such that*

 ∞ *a*

Φ(*t*)*d f*(*t*)


*<sup>a</sup> ϕdu is bounded on* [*a*, ∞). *Then ϕ f* ∈ *HK*([*a*, *b*]),

*ϕ f dt* = −

 ∞ *a*

*ϕ f dt* ≤ sup *a<t*

**Theorem 12.** *[18, Theorem 4] Let a*, *b* ∈ **R***. If h* : **R** × [*a*, *b*] → **C** *is such that*

<sup>−</sup><sup>∞</sup> *<sup>h</sup>*(*t*, ·)*dt belongs to ACG<sup>δ</sup> on* [*a*, *<sup>b</sup>*] *and H*

*D*2*h*(*t*,*s*)*dtds* =

 ∞ −∞  *t s*

 ∞ *a*

*Similar results are valid for the cases* [−∞, ∞] *and* [−∞, *a*].

there exist *ηε <sup>&</sup>gt;* 0 and a gauge *δε* on *<sup>E</sup>* such that if {(*xi*, *yi*)}*<sup>N</sup>*

*<sup>i</sup>* (*yi* <sup>−</sup> *xi*) *<sup>&</sup>lt; ηε*, then <sup>∑</sup>*<sup>N</sup>*

to the class *ACGδ*(*I*) if there exists a sequence {*En*}<sup>∞</sup>

*1. h*(*t*, ·) *belongs to ACG<sup>δ</sup> on* [*a*, *b*] *for almost all t* ∈ **R***;*

 *t s*

*2. and h*(·,*s*) *is a HK integrable function on* **R** *for all s* ∈ [*a*, *b*]*.*

 ∞ −∞ *b*

*<sup>a</sup>*+*<sup>ε</sup> ϕ*(*t*)*dt exists. In this case, this*

*<sup>i</sup>* is a (*δε*, *E*)−fine subpartition

<sup>−</sup><sup>∞</sup> *<sup>D</sup>*2*h*(*t*,*s*)*dt for almost all*

*<sup>n</sup>*=1*En* and

<sup>1</sup> |*F*(*xi*) − *F*(*yi*)| *<* . On the other hand, *F* belongs

(*s*) = <sup>∞</sup>

*D*2*h*(*t*,*s*)*dsdt*

<sup>1</sup> of sets in *<sup>I</sup>* such that *<sup>I</sup>* <sup>=</sup> <sup>∪</sup><sup>∞</sup>

*<sup>a</sup> g is bounded on* [*a*, ∞)*.*

*<sup>b</sup>* <sup>−</sup> *<sup>a</sup> <sup>&</sup>gt; <sup>ε</sup> <sup>&</sup>gt;* 0, *it follows that <sup>ϕ</sup>* <sup>∈</sup> *HK*([*<sup>a</sup>* <sup>+</sup> *<sup>ε</sup>*, *<sup>b</sup>*]) *and* lim *<sup>ε</sup>*→0, *<sup>b</sup>*→<sup>∞</sup>

*1. <sup>g</sup>* <sup>∈</sup> *HK*([*a*, *<sup>c</sup>*]) *for every c* <sup>≥</sup> *a, and G defined by G*(*x*) = *<sup>x</sup>*

*2. f is of bounded variation on* [*a*, <sup>∞</sup>) *and* lim*x*→<sup>∞</sup> *<sup>f</sup>*(*x*) = <sup>0</sup>*.*

$$H'(s\_0) = \int\_{-\infty}^{\infty} D\_2 h(t, s\_0) dt$$

*when H*<sup>2</sup> :<sup>=</sup> <sup>∞</sup> <sup>−</sup><sup>∞</sup> *<sup>D</sup>*2*h*(*t*, ·)*dt is continuous at s*0*.*

## **2. Fourier coefficients for functions in the Henstock-Kurzweil completion.**

For finite intervals, the Theorem 12.11 of [19] tells us that: In order that *<sup>b</sup> <sup>a</sup> f gn* → *b <sup>a</sup> f g*, *n* → ∞, whenever *f* ∈ *HK*([*a*, *b*]), it is necessary and sufficient that: i) *gn* is almost everywhere of bounded variation on [*a*, *b*] for each *n*; ii) sup{||*gn*||<sup>∞</sup> + ||*gn*||*BV*} *<* ∞; iii) *<sup>d</sup> <sup>c</sup> gn* <sup>→</sup> *<sup>d</sup> <sup>c</sup> g*, *n* → ∞, for each interval (*c*, *d*) ⊂ (*a*, *b*). The Theorem 3 of [8] proves that above theorem is valid for infinite intervals. In this section we show that [19, Theorem 12.11] and [8, Theorem 3] are true for functions belonging to the completion of the Henstock-Kurzweil space. First, we need to prove the next lemma. The class of step functions on [*a*, *b*] will be denoted as *K*([*a*, *b*]).

**Lemma 13.** *Let* [*a*, *b*] *be an infinite interval. The set K*([*a*, *b*]) *is dense in HK*([*a*, *b*])*.*

*Proof.* Let *f* ∈ *HK*([*a*, ∞]) and *>* 0 be given. By Hake's Theorem, exists *N* ∈ **N** such that for each *x* ≥ *N*,

$$\left| \int\_{x}^{\infty} f \right| < \frac{\varepsilon}{2}. \tag{8}$$

Since *K*([*a*, *N*]) is dense in *HK*([*a*, *N*]), by Theorem 7 of [20], there exists a function *h* ∈ *K*([*a*, *N*]) such that

$$||f - h||\_{A, [a, N]} = \sup\_{\mathbf{x} \in [a, N]} \left| \int\_{a}^{\mathbf{x}} (f - h) \right| < \frac{\epsilon}{2}. \tag{9}$$

Defining *h*<sup>0</sup> ∈ *K*([*a*, ∞]) as

$$h\_0(\mathbf{x}) = \begin{cases} h(\mathbf{x}) \text{ if } \mathbf{x} \in [a\_\prime N] \\ 0 \text{ if } \mathbf{x} \in (N\_\prime \infty] .\end{cases}$$

It follows, by (8) and (9), that

$$||f - h\_0||\_A \le \epsilon.$$

Similar arguments apply for intervals as [−∞, *a*] or [−∞, ∞].

#### **2.1. The convergence of integrals of products in the completion**

The following result appears in [3]. Here, we present a detailed proof.

**Theorem 14.** *Let* [*a*, *b*] ⊂ **R***. In order that*

$$\int\_{a}^{b} f \mathbf{g}\_{n} \to \int\_{a}^{b} f \mathbf{g}\_{\prime} \quad n \to \infty,\tag{10}$$

*whenever f* <sup>∈</sup> *HK*([*a*, *<sup>b</sup>*])*, it is necessary and sufficient that: i) gn is almost everywhere of bounded variation on* [*a*, *<sup>b</sup>*] *for each n; ii*) sup{||*gn*||<sup>∞</sup> <sup>+</sup> ||*gn*||*BV*} *<sup>&</sup>lt;* <sup>∞</sup>*; iii) <sup>d</sup> <sup>c</sup> gn* <sup>→</sup> *<sup>d</sup> <sup>c</sup> g, n* → ∞, *for each interval* (*c*, *d*) ⊂ (*a*, *b*).

*Proof.* The necessity follows from [19, Theorem 12.11]. Now we will prove the sufficiency condition. Define the linear functionals *<sup>T</sup>*, *Tn* : *HK*([*a*, *<sup>b</sup>*]) <sup>→</sup> **<sup>R</sup>** by

$$T\_{\hbar}(f) = \int\_{a}^{b} f \mathbf{g}\_{\hbar} \quad \text{and} \quad T(f) = \int\_{a}^{b} f \mathbf{g}. \tag{11}$$

Supposing *i*) and *ii*), we have, by Multiplier Theorem, that the sequence {*Tn*} is bounded by sup{||*gn*||<sup>∞</sup> + ||*gn*||*BV*}. Owing to Lemma 13, the space of step functions is dense in *HK*([*a*, *<sup>b</sup>*]), then considering the Theorem <sup>6</sup> it is sufficient to prove that {*Tn*(*f*)} converge to *T*(*f*), for each step function *f* . First, let *f*(*x*) = *χ*(*c*,*d*)(*x*) be the characteristic function of (*c*, *d*) ⊂ [*a*, *b*]. Thus,

$$T\_n(f) = \int\_a^b \chi\_{(c,d)} g\_n = \int\_c^d g\_{n\nu}$$

by the hypothesis *iii*), we have that {*Tn*(*χ*(*c*,*d*))} converges to *T*(*χ*(*c*,*d*)), as *n* → ∞. Now, let *f* be a step function. Being that each *Tn* is a linear functional, then {*Tn*(*f*)} converges to *T*(*f*), as *n* → ∞. Thus, the result holds.

**Remark 15.** *On HK*([*a*, *b*])*. The hypothesis iii*) *can be replaced by: gn converges pointwise to g*, *then the result follows from Corollary 3.2 of [21]. The result on the completion holds by Theorem 6. Note that the conditions i*)*, ii*) *and iii*) *do not imply converges pointwise from* {*gn*} *to g*, *see example 2 of [8].*

For the case of functions defined on a finite interval we get Theorem 16, and a lemma of Riemann-Lebesgue type for functions in the Henstock-Kurzweil space completion.

**Theorem 16.** *Let* [*a*, *b*] *be a finite interval. If i) gn converges to g in measure on* [*a*, *b*]*, ii) each gn is equal to hn almost everywhere, a normalized bounded variation function and iii*) *there is M >* 0 *such that Var*(*hn*, [*a*, *<sup>b</sup>*]) <sup>≤</sup> *M, n* <sup>≥</sup> <sup>1</sup>*, then for all f* <sup>∈</sup> *HK*([*a*, *<sup>b</sup>*])*,*

$$\int\_{a}^{b} f \mathbf{g}\_{n} \to \int\_{a}^{b} f \mathbf{g}\_{\prime} \quad n \to \infty.$$

*Proof.* Let *<sup>f</sup>* <sup>∈</sup> *HK*([*a*, *<sup>b</sup>*]) be given, where *<sup>f</sup>* = [{ *fk*}], we want to prove that

$$\lim\_{n \to \infty} \lim\_{k \to \infty} \int\_a^b f\_k g\_n = \int\_a^b f g.$$

Define *f*(*k*, *n*) = *<sup>b</sup> <sup>a</sup> fkgn*. By the hypothesis *i*) about (*gn*) we have

$$\lim\_{n \to \infty} \int\_{a}^{b} f\_{k} \mathbf{g}\_{n} = \int\_{a}^{b} f\_{k} \mathbf{g}\_{n}$$

Moreover

8 ime knjige

**2.1. The convergence of integrals of products in the completion** The following result appears in [3]. Here, we present a detailed proof.

> *b a*

*variation on* [*a*, *<sup>b</sup>*] *for each n; ii*) sup{||*gn*||<sup>∞</sup> <sup>+</sup> ||*gn*||*BV*} *<sup>&</sup>lt;* <sup>∞</sup>*; iii) <sup>d</sup>*

condition. Define the linear functionals *<sup>T</sup>*, *Tn* : *HK*([*a*, *<sup>b</sup>*]) <sup>→</sup> **<sup>R</sup>** by

*Tn*(*f*) =

 *b a*

*Tn*(*f*) =

 *b a*

*f gn* →

 *b a*

*whenever f* <sup>∈</sup> *HK*([*a*, *<sup>b</sup>*])*, it is necessary and sufficient that: i) gn is almost everywhere of bounded*

*Proof.* The necessity follows from [19, Theorem 12.11]. Now we will prove the sufficiency

*f gn* and *T*(*f*) =

*χ*(*c*,*d*)*gn* =

by the hypothesis *iii*), we have that {*Tn*(*χ*(*c*,*d*))} converges to *T*(*χ*(*c*,*d*)), as *n* → ∞. Now, let *f* be a step function. Being that each *Tn* is a linear functional, then {*Tn*(*f*)} converges to *T*(*f*),

**Remark 15.** *On HK*([*a*, *b*])*. The hypothesis iii*) *can be replaced by: gn converges pointwise to g*, *then the result follows from Corollary 3.2 of [21]. The result on the completion holds by Theorem 6. Note that the conditions i*)*, ii*) *and iii*) *do not imply converges pointwise from* {*gn*} *to g*, *see example*

For the case of functions defined on a finite interval we get Theorem 16, and a lemma of

**Theorem 16.** *Let* [*a*, *b*] *be a finite interval. If i) gn converges to g in measure on* [*a*, *b*]*, ii) each gn is equal to hn almost everywhere, a normalized bounded variation function and iii*) *there is M >* 0 *such*

> *b a*

*f g*, *n* → ∞.

Riemann-Lebesgue type for functions in the Henstock-Kurzweil space completion.

*f gn* →

*that Var*(*hn*, [*a*, *<sup>b</sup>*]) <sup>≤</sup> *M, n* <sup>≥</sup> <sup>1</sup>*, then for all f* <sup>∈</sup> *HK*([*a*, *<sup>b</sup>*])*,*

 *b a*

 *d c gn*,

Supposing *i*) and *ii*), we have, by Multiplier Theorem, that the sequence {*Tn*} is bounded by sup{||*gn*||<sup>∞</sup> + ||*gn*||*BV*}. Owing to Lemma 13, the space of step functions is dense in *HK*([*a*, *<sup>b</sup>*]), then considering the Theorem <sup>6</sup> it is sufficient to prove that {*Tn*(*f*)} converge to *T*(*f*), for each step function *f* . First, let *f*(*x*) = *χ*(*c*,*d*)(*x*) be the characteristic function of

 *b a*

*f g*, *n* → ∞, (10)

*<sup>c</sup> gn* <sup>→</sup> *<sup>d</sup>*

*f g*. (11)

*<sup>c</sup> g, n* → ∞, *for*

**Theorem 14.** *Let* [*a*, *b*] ⊂ **R***. In order that*

*each interval* (*c*, *d*) ⊂ (*a*, *b*).

(*c*, *d*) ⊂ [*a*, *b*]. Thus,

*2 of [8].*

as *n* → ∞. Thus, the result holds.

$$\lim\_{k \to \infty} \lim\_{n \to \infty} \int\_a^b f\_k g\_n = \int\_a^b f g\_n$$

by the integral definition on the completion. We will prove that lim*k*→<sup>∞</sup> *<sup>f</sup>*(*k*, *<sup>n</sup>*) = *<sup>b</sup> <sup>a</sup> f gn* converges uniformly on *n*. Let *>* 0 be given, there exists *k*<sup>0</sup> such that || *fk* − *f* ||*<sup>A</sup>* ≤ , if *k* ≥ *k*0. Besides, if *k* ≥ *k*0,

$$\begin{aligned} \left| \int\_a^b f\_k \mathfrak{g}\_n - \int\_a^b f \mathfrak{g}\_n \right| &\leq ||f\_k - f||\_A \mathrm{Var}(\mathfrak{g}\_n, [a, b])| \\ &\leq M \varepsilon. \end{aligned}$$

Therefore, by Theorem 5,

$$\lim\_{n \to \infty} \lim\_{k \to \infty} \int\_{a}^{b} f\_{k} \mathbf{g}\_{n} = \int\_{a}^{b} f \mathbf{g}.$$

The following result is a "generalization" of Riemann-Lebesgue Lemma on the completion of the space *HK*([*a*, *b*]), over finite intervals, it also appears in [3].

**Corollary 17.** *If ϕ* : **R** → **R** *such that ϕ exists, is bounded and ϕ*(*s*) = *o*(*s*), *as* |*s*| → ∞*, then for each f* <sup>∈</sup> *HK*([*a*, *<sup>b</sup>*]) *we have the next asymptotic behavior*

$$\int\_{a}^{b} \varrho(st)f(t)dt = o(s), \quad as \quad |s| \to \infty.$$

*Proof.* For each *s* = 0 define *ϕ<sup>s</sup>* : **R** → **R** as *ϕs*(*t*) = *ϕ*(*st*)/*s*. In order to prove

$$\lim\_{s \to \infty} \int\_a^b \frac{\varrho(st)}{s} f(t) dt = 0\_r$$

it is sufficient to show that *ϕ<sup>s</sup>* fulfills the hypothesis of Theorem 16. Now, we will check item by item. *i*) Because of *ϕ*(*s*) = *o*(*s*), as |*s*| → ∞ and the interval [*a*, *b*] is finite, it follows that *ϕ<sup>s</sup>* converges in measure to 0. *ii*) Owing to *ϕ* is bounded then, by the Mean Value Theorem, we have that *ϕ<sup>s</sup>* ∈ *BV*([*a*, *b*]). *iii*) *Var*(*ϕs*, [*a*, *b*]) is bounded uniformly by upper bound of *ϕ* and *a* − *b*.

#### **2.2. Riemann-Lebesgue Property**

This property establishes that *<sup>π</sup> <sup>r</sup> <sup>f</sup>*(*t*)*Dn*(*t*)*dt* <sup>→</sup> 0, for each *<sup>f</sup>* <sup>∈</sup> *<sup>L</sup>*1[−*π*, *<sup>π</sup>*] and *<sup>r</sup>* <sup>∈</sup> (0, *<sup>π</sup>*], where *Dn*(*t*) = sin(*n*+1/2)*<sup>t</sup>* sin(*t*/2) denotes the n-th Dirichlet Kernel of order *n*. Now, we provide an analogous result concerning the Henstock-Kurzweil completion.

**Theorem 18.** *For any f* <sup>∈</sup> *HK*([<sup>−</sup>*π*, *<sup>π</sup>*])*, and r* <sup>∈</sup> (0, *<sup>π</sup>*]*,*

$$\lim\_{n \to \infty} \frac{1}{n} \int\_{r}^{\pi} f(t) D\_{n}(t) dt = 0. \tag{12}$$

*Proof.* Note that the function *g*(*t*) = 1/ sin(*t*/2) is in *BV*([*r*, *π*]). Moreover, by Multiplier Theorem we have *f g* <sup>∈</sup> *HK*([*r*, *<sup>π</sup>*]). Hence, by Corollary 17, we get

$$\int\_{r}^{\pi} f(t)g(t)\sin(n+1/2)t dt = o(n), \quad |n| \to \infty.$$
 
$$\square$$

Considering an similar argument from above proof, it follows that

$$\int\_{\pi}^{\pi} f(t) \frac{\sin(n + 1/2)t}{t/2} dt = o(n), \quad |n| \to \infty. \tag{13}$$

For *<sup>n</sup>* <sup>∈</sup> **<sup>N</sup>** ∪ {0}, we define the function <sup>Φ</sup>*n*(*t*) = sin(*n*+1/2)*<sup>t</sup> <sup>t</sup>*/2 for *t* = 0 and Φ*n*(0) = 2*n* +1, it is called the discrete Fourier Kernel of order *n*. This kernel provides a very good approximation to the Dirichlet Kernel *Dn* for |*t*| *<* 2, but Φ*<sup>n</sup>* decreases more rapidly than *Dn*, see [1].

**Theorem 19.** *Let f* <sup>∈</sup> *HK*([0, *<sup>π</sup>*]) *and r* <sup>∈</sup> (0, *<sup>π</sup>*]*. Then, assuming that any of next limits exist,*

$$\lim\_{n \to \infty} \frac{1}{n} \int\_0^r f(t) D\_n(t) dt = \lim\_{n \to \infty} \frac{1}{n} \int\_0^r f(t) \frac{\sin(n + 1/2)t}{t/2} dt.$$

*Proof.* Define *g* : [0, *π*] → **R** by

$$g(t) = \begin{cases} \frac{1}{\sin(t/2)} - \frac{1}{t/2} & \text{for } t \in (0, \pi] \\\\ 0 & \text{for} \quad t = 0. \end{cases}$$

Since *<sup>g</sup>* <sup>∈</sup> *BV*([0, *<sup>π</sup>*]), *f g* <sup>∈</sup> *HK*([0, *<sup>π</sup>*]). By Corollary 17, we have

$$\lim\_{n \to \infty} \frac{1}{n} \int\_0^\pi f(t) \left( \frac{1}{\sin(t/2)} - \frac{1}{t/2} \right) \sin \left( n + 1/2 \right) t dt = 0. \tag{14}$$

Now, by (14), we have

$$\lim\_{n \to \infty} \frac{1}{n} \int\_0^\pi \left( f(t) D\_n(t) - f(t) \frac{\sin(n + 1/2)t}{t/2} \right) dt = 0.$$

Then

10 ime knjige

**2.2. Riemann-Lebesgue Property** This property establishes that *<sup>π</sup>*

analogous result concerning the Henstock-Kurzweil completion.

lim*n*→<sup>∞</sup> 1 *n π r*

Theorem we have *f g* <sup>∈</sup> *HK*([*r*, *<sup>π</sup>*]). Hence, by Corollary 17, we get

Considering an similar argument from above proof, it follows that

sin(*n* + 1/2)*t*

*f*(*t*)

For *<sup>n</sup>* <sup>∈</sup> **<sup>N</sup>** ∪ {0}, we define the function <sup>Φ</sup>*n*(*t*) = sin(*n*+1/2)*<sup>t</sup>*

*g*(*t*) =

Since *<sup>g</sup>* <sup>∈</sup> *BV*([0, *<sup>π</sup>*]), *f g* <sup>∈</sup> *HK*([0, *<sup>π</sup>*]). By Corollary 17, we have

 *π r*

> *π r*

lim*n*→<sup>∞</sup> 1 *n r* 0

*Proof.* Define *g* : [0, *π*] → **R** by

<sup>−</sup>*π*, *<sup>π</sup>*])*, and r* <sup>∈</sup> (0, *<sup>π</sup>*]*,*

*Proof.* Note that the function *g*(*t*) = 1/ sin(*t*/2) is in *BV*([*r*, *π*]). Moreover, by Multiplier

*f*(*t*)*g*(*t*) sin(*n* + 1/2)*tdt* = *o*(*n*), |*n*| → ∞.

called the discrete Fourier Kernel of order *n*. This kernel provides a very good approximation to the Dirichlet Kernel *Dn* for |*t*| *<* 2, but Φ*<sup>n</sup>* decreases more rapidly than *Dn*, see [1].

> 1 *n r* 0 *f*(*t*)

*<sup>t</sup>*/2 for *t* ∈ (0, *π*]

0 for *t* = 0.

**Theorem 19.** *Let f* <sup>∈</sup> *HK*([0, *<sup>π</sup>*]) *and r* <sup>∈</sup> (0, *<sup>π</sup>*]*. Then, assuming that any of next limits exist,*

*<sup>f</sup>*(*t*)*Dn*(*t*)*dt* <sup>=</sup> lim*n*→<sup>∞</sup>

 

1 sin(*t*/2) <sup>−</sup> <sup>1</sup>

where *Dn*(*t*) = sin(*n*+1/2)*<sup>t</sup>*

**Theorem 18.** *For any f* ∈ *HK*([

*<sup>r</sup> <sup>f</sup>*(*t*)*Dn*(*t*)*dt* <sup>→</sup> 0, for each *<sup>f</sup>* <sup>∈</sup> *<sup>L</sup>*1[−*π*, *<sup>π</sup>*] and *<sup>r</sup>* <sup>∈</sup> (0, *<sup>π</sup>*],

*f*(*t*)*Dn*(*t*)*dt* = 0. (12)

*<sup>t</sup>*/2 *dt* <sup>=</sup> *<sup>o</sup>*(*n*), <sup>|</sup>*n*| → <sup>∞</sup>. (13)

sin(*n* + 1/2)*t <sup>t</sup>*/2 *dt*.

*<sup>t</sup>*/2 for *t* = 0 and Φ*n*(0) = 2*n* +1, it is

sin(*t*/2) denotes the n-th Dirichlet Kernel of order *n*. Now, we provide an

$$\begin{aligned} \lim\_{n \to \infty} \frac{1}{n} \left[ \int\_0^r \left( f(t) D\_n(t) - f(t) \frac{\sin(n + 1/2)t}{t/2} \right) dt \\ &+ \int\_r^\pi \left( f(t) D\_n(t) - f(t) \frac{\sin(n + 1/2)t}{t/2} \right) dt \right] = 0. \end{aligned}$$

By Theorem 18 and (13),

$$\lim\_{n \to \infty} \frac{1}{n} \int\_{r}^{\pi} \left( f(t) D\_n(t) - f(t) \frac{\sin(n + 1/2)t}{t/2} \right) dt = 0.$$

Therefore, assuming that any of the limits exist, we have

$$\lim\_{n \to \infty} \frac{1}{n} \int\_0^r f(t) D\_n(t) dt = \lim\_{n \to \infty} \frac{1}{n} \int\_0^r f(t) \frac{\sin(n + 1/2)t}{t/2} dt.$$

The following result is a characterization of the asymptotic behavior of *n* − *th* partial sum of the Fourier series, it can be found in [3].

**Corollary 20.** *Let f* <sup>∈</sup> *HK*([−*π*, *<sup>π</sup>*]) *be* <sup>2</sup>*π*<sup>−</sup> *periodic. The n* <sup>−</sup> *th partial sum of the Fourier series at t has the following asymptotic behavior Sn*(*f* , *t*) = *o*(*n*)*, when* |*n*| → ∞ *iff*

$$\int\_0^\pi [f(t+u) + f(t-u)] \frac{\sin(n+1/2)u}{u} du = o(n)\_\nu$$

*if* |*n*| → ∞*.*

*Proof.* Since *Sn*(*<sup>f</sup>* , *<sup>t</sup>*) = *<sup>π</sup>* <sup>−</sup>*<sup>π</sup> <sup>f</sup>*(*<sup>t</sup>* <sup>+</sup> *<sup>u</sup>*)*Dn*(*u*)*du*, realizing a change of variable (see section 6 of [13]), then by Theorem 19 we get the result.

## **3. Henstock-Kurzweil integral transform**

The results in this section are based for functions in the vector space *BV*0(**R**), and they have to [10] as principal reference.

We will introduce some additional terminology in order to facilitate the following results.

If *g* : **R** × **R** → **C** is a function and *s*<sup>0</sup> ∈ **R**, we say that *s*<sup>0</sup> fulfills hypothesis (**H**) relative to *g* if:

(**H**) there exist *δ* = *δ*(*s*0) *>* 0 and *M* = *M*(*s*0) *>* 0, such that, if |*s* − *s*0| *< δ* then

$$\left| \int\_{u}^{v} g(t, s) dt \right| \leq M\_{\prime}$$

for all [*u*, *v*] ⊆ **R**.

This condition plays a significant role in the following results. Also, the next theorems can be found in [10].

**Theorem 21.** *Let f* : **R** → **R** *and g* : **R** × **R** → **C** *be functions. Assume that f* ∈ *BV*0(**R**)*, and s*<sup>0</sup> ∈ **R** *fulfills Hypothesis* (**H**) *relative to g, then*

$$
\Gamma(s) = \int\_{-\infty}^{\infty} f(t)g(t,s)dt
$$

*is well defined for all s in a neighborhood of s*0*.*

*Proof.* Applying Theorem 9 the result holds.

**Theorem 22.** *Let f* : **R** → **R** *and g* : **R** × **R** → **C** *be functions assume that*


*If s*<sup>0</sup> ∈ **R** *fulfills Hypothesis* (**H**) *relative to g, then the function* Γ *is continuous at s*0*.*

*Proof.* By Hypothesis (**H**), there exist *δ*<sup>1</sup> *>* 0 and *M >* 0, such that, if |*s* − *s*0| *< δ*<sup>1</sup> then

$$\left| \int\_{u}^{v} \mathbf{g}(t, \mathbf{s}) dt \right| \leq M \tag{15}$$

for all [*u*, *v*] ⊆ **R**. From Theorem 21, Γ(*s*) exists for all *s* ∈ *Bδ*<sup>1</sup> (*s*0).

Let an arbitrary *>* 0, by Hake's theorem, there exists *K*<sup>1</sup> *>* 0 such that

$$\left| \int\_{|t| \ge \underline{u}} f(t)g(t, s\_0)dt \right| < \frac{\varepsilon}{3} \tag{16}$$

for all *u* ≥ *K*1. On the other hand, as

12 ime knjige

if:

**3. Henstock-Kurzweil integral transform**

to [10] as principal reference.

for all [*u*, *v*] ⊆ **R**.

*s*<sup>0</sup> ∈ **R** *fulfills Hypothesis* (**H**) *relative to g, then*

*is well defined for all s in a neighborhood of s*0*.*

*Proof.* Applying Theorem 9 the result holds.

*1. f belongs to BV*0(**R**)*, g is bounded, and*

*2. g*(*t*, ·) *is continuous for all t* ∈ **R***.*

be found in [10].

The results in this section are based for functions in the vector space *BV*0(**R**), and they have

We will introduce some additional terminology in order to facilitate the following results.

(**H**) there exist *δ* = *δ*(*s*0) *>* 0 and *M* = *M*(*s*0) *>* 0, such that, if |*s* − *s*0| *< δ* then

 *v u*

Γ(*s*) =

**Theorem 22.** *Let f* : **R** → **R** *and g* : **R** × **R** → **C** *be functions assume that*

*If s*<sup>0</sup> ∈ **R** *fulfills Hypothesis* (**H**) *relative to g, then the function* Γ *is continuous at s*0*.*

 *v u*

for all [*u*, *v*] ⊆ **R**. From Theorem 21, Γ(*s*) exists for all *s* ∈ *Bδ*<sup>1</sup> (*s*0).

Let an arbitrary *>* 0, by Hake's theorem, there exists *K*<sup>1</sup> *>* 0 such that

*Proof.* By Hypothesis (**H**), there exist *δ*<sup>1</sup> *>* 0 and *M >* 0, such that, if |*s* − *s*0| *< δ*<sup>1</sup> then

*g*(*t*,*s*)*dt*

 

≤ *M* (15)

If *g* : **R** × **R** → **C** is a function and *s*<sup>0</sup> ∈ **R**, we say that *s*<sup>0</sup> fulfills hypothesis (**H**) relative to *g*

*g*(*t*,*s*)*dt*

This condition plays a significant role in the following results. Also, the next theorems can

**Theorem 21.** *Let f* : **R** → **R** *and g* : **R** × **R** → **C** *be functions. Assume that f* ∈ *BV*0(**R**)*, and*

 ∞ −∞  ≤ *M*,

*f*(*t*)*g*(*t*,*s*)*dt*

$$\lim\_{t \to -\infty} \text{Var}(f\_{\prime}(-\infty, t]) = 0 \quad \text{and} \quad \lim\_{t \to \infty} \text{Var}(f\_{\prime}(t, \infty)) = 0\_{\prime \nu}$$

there is *K*<sup>2</sup> *>* 0 such that for each *t > K*2,

$$\text{Var}(f\_{\prime}(-\infty,-t]) + \text{Var}(f\_{\prime}(t,\infty)) < \frac{\epsilon}{3M}.$$

Let *K* = max{*K*1, *K*2}. From Theorem 7, it follows that for every *v* ≥ *K* and every *s* ∈ *Bδ*<sup>1</sup> (*s*0),

$$\begin{aligned} \left| \int\_{K}^{v} f(t)g(t,s)dt \right| &\leq \|g(\cdot,s)\|\_{[K,v]} \left[ \inf\_{t \in [K,v]} |f(t)| + \text{Var}(f,[K,v]) \right] \\ &\leq M \left[ |f(v)| + \text{Var}(f,[K,\infty)) \right] .\end{aligned}$$

where the second inequality is true due to (15). This implies, since lim*t*→<sup>∞</sup> | *f*(*t*)| = 0, that

$$\left| \int\_{K}^{\infty} f(t)g(t,s)dt \right| \leq M \cdot \text{Var}(f, [K, \infty)).$$

Analogously we have that

$$\left| \int\_{-\infty}^{-K} f(t)g(t,s)dt \right| \le M \cdot \text{Var}(f, (-\infty, -K]).$$

Therefore, for each *s* ∈ *Bδ*<sup>1</sup> (*s*0),

$$\left| \int\_{|t| \ge K} f(t)g(t,s)dt \right| \le M \left[ \text{Var}(f\_{\prime}(-\infty, -K])f + \text{Var}(f\_{\prime}[K, \infty)) \right]$$

$$ < M \frac{\mathfrak{e}}{3M} = \frac{\mathfrak{e}}{3}. \tag{17}$$

Since *<sup>f</sup>* is *<sup>L</sup>*1[−*K*, *<sup>K</sup>*], *<sup>g</sup>* is bounded and *<sup>g</sup>*(*t*, ·) is continuous for all *<sup>t</sup>* <sup>∈</sup> **<sup>R</sup>**. For example, using Theorem 12.12 of [11], it is easy to show that the function

$$\Gamma\_K(s) = \int\_{-K}^{K} f(t)g(t,s)dt, \text{ s} \in \mathbb{R}\_{\text{-}}$$

is continuous at *s*0. This implies that there is *δ*<sup>2</sup> *>* 0 such that for every *s* ∈ *Bδ*<sup>2</sup> (*s*0),

$$\left| \int\_{-K}^{K} f(t) [g(t, s) - g(t, s\_0)] dt \right| < \frac{\epsilon}{3}. \tag{18}$$

Let *δ* = min{*δ*1, *δ*2}. Then for all *s* ∈ *Bδ*(*s*0),

$$\begin{split} |\Gamma(\mathbf{s}) - \Gamma(\mathbf{s}\_0)| &\leq \left| \int\_{-K}^{K} f(t) [\mathbf{g}(t, \mathbf{s}) - \mathbf{g}(t, \mathbf{s}\_0)] dt \right| \\ &+ \left| \int\_{|t| \geq K} f(t) \mathbf{g}(t, \mathbf{s}) dt \right| + \left| \int\_{|t| \geq K} f(t) \mathbf{g}(t, \mathbf{s}\_0) dt \right|. \end{split}$$

Thus, from (16), (17) and (18), <sup>|</sup>Γ(*s*) <sup>−</sup> <sup>Γ</sup>(*s*0)<sup>|</sup> *<sup>&</sup>lt;* <sup>3</sup> <sup>+</sup> <sup>3</sup> <sup>+</sup> <sup>3</sup> = , for all *s* ∈ *Bδ*(*s*0).

**Theorem 23.** *Let a*, *b* ∈ **R***. If f* : **R** → **R** *and g* : **R** × [*a*, *b*] → **C** *are functions such that*


*Then*

$$\int\_{a}^{b} \int\_{-\infty}^{\infty} f(t)g(t,s)dtds = \int\_{-\infty}^{\infty} \int\_{a}^{b} f(t)g(t,s)dsdt$$

*Proof.* From (2) and since [*a*, *b*] is compact, there exists *M >* 0 such that, for every *s* ∈ [*a*, *b*] and for all [*u*, *v*] ⊆ **R** : *v <sup>u</sup> <sup>g</sup>*(*t*,*s*)*dt* ≤ *M*.

For *<sup>r</sup> <sup>&</sup>gt;* 0 and *<sup>s</sup>* <sup>∈</sup> [*a*, *<sup>b</sup>*], let <sup>Γ</sup>*r*(*s*) = *<sup>r</sup>* <sup>−</sup>*<sup>r</sup> <sup>f</sup>*(*t*)*g*(*t*,*s*)*dt*. By Theorem 7, we notice that

$$\begin{aligned} |\Gamma\_l(s)| &= \left| \int\_{-r}^r f(t)g(t,s)dt \right| \\ &\le ||g(\cdot,s)||\_{[-r,r]} \left[ \inf\_{t \in [-r,r]} |f(t)| + V\_{[-r,r]}f \right] \\ &\le M[|f(0)| + Vf] \end{aligned}$$

for all *s* ∈ [*a*, *b*].

So, for each *r >* 0, Γ*<sup>r</sup>* is HK integrable on [*a*, *b*] and is bounded for a fixed constant. Moreover, by Theorem 21 and Hake's theorem

$$\lim\_{r \to \infty} \Gamma\_r(s) = \Gamma(s)$$

for all *s* ∈ [*a*, *b*].

14 ime knjige

*Then*

and for all [*u*, *v*] ⊆ **R** :

for all *s* ∈ [*a*, *b*].

Since *<sup>f</sup>* is *<sup>L</sup>*1[−*K*, *<sup>K</sup>*], *<sup>g</sup>* is bounded and *<sup>g</sup>*(*t*, ·) is continuous for all *<sup>t</sup>* <sup>∈</sup> **<sup>R</sup>**. For example, using

*f*(*t*)[*g*(*t*,*s*) − *g*(*t*,*s*0)]*dt*

*f*(*t*)[*g*(*t*,*s*) − *g*(*t*,*s*0)]*dt*

*f*(*t*)*g*(*t*,*s*)*dt*

<sup>3</sup> <sup>+</sup> <sup>3</sup> <sup>+</sup>

 ∞ −∞

*Proof.* From (2) and since [*a*, *b*] is compact, there exists *M >* 0 such that, for every *s* ∈ [*a*, *b*]

 

inf *t*∈[−*r*,*r*]

*f*(*t*)*g*(*t*,*s*)*dt*

 *b a*

 + 

*f*(*t*)*g*(*t*,*s*)*dt*, *s* ∈ **R**,

 *<* 3

> 


*f*(*t*)*g*(*t*,*s*)*dsdt*

<sup>−</sup>*<sup>r</sup> <sup>f</sup>*(*t*)*g*(*t*,*s*)*dt*. By Theorem 7, we notice that

<sup>|</sup> *<sup>f</sup>*(*t*)<sup>|</sup> <sup>+</sup> *<sup>V</sup>*[−*r*,*r*] *<sup>f</sup>*

*f*(*t*)*g*(*t*,*s*0)*dt*

<sup>3</sup> = , for all *s* ∈ *Bδ*(*s*0).

. (18)

 .

Theorem 12.12 of [11], it is easy to show that the function

 *K* −*K*

> *K* −*K*

+ 


**Theorem 23.** *Let a*, *b* ∈ **R***. If f* : **R** → **R** *and g* : **R** × [*a*, *b*] → **C** *are functions such that*

*f*(*t*)*g*(*t*,*s*)*dtds* =

≤ *M*.

≤ *g*(·,*s*)[−*r*,*r*]

≤ *M*[| *f*(0)| + *V f* ]

Let *δ* = min{*δ*1, *δ*2}. Then for all *s* ∈ *Bδ*(*s*0),


Thus, from (16), (17) and (18), <sup>|</sup>Γ(*s*) <sup>−</sup> <sup>Γ</sup>(*s*0)<sup>|</sup> *<sup>&</sup>lt;*

 *b a*

 *v*


For *<sup>r</sup> <sup>&</sup>gt;* 0 and *<sup>s</sup>* <sup>∈</sup> [*a*, *<sup>b</sup>*], let <sup>Γ</sup>*r*(*s*) = *<sup>r</sup>*

*2. for all s* ∈ [*a*, *b*]*, s satisfies Hypothesis* (**H**) *relative to g.*

 ∞ −∞

*<sup>u</sup> <sup>g</sup>*(*t*,*s*)*dt*

 *r* −*r*

*1. f* ∈ *BV*0(**R**)*, g is measurable, bounded and*

Γ*K*(*s*) =

 *K* −*K*

is continuous at *s*0. This implies that there is *δ*<sup>2</sup> *>* 0 such that for every *s* ∈ *Bδ*<sup>2</sup> (*s*0),

Using the Lebesgue Dominated Convergence Theorem, we have that Γ is HK integrable on [*a*, *b*] and

$$\int\_{a}^{b} \Gamma(s)ds = \lim\_{r \to \infty} \int\_{a}^{b} \Gamma\_{r}(s)ds.$$

Now, because of *f* is Lebesgue integrable on [−*r*, *r*]; *g* is measurable and bounded; and by Fubini's theorem, it follows that

$$\int\_{a}^{b} \int\_{-r}^{r} f(t)g(t,s)dtds = \int\_{-r}^{r} \int\_{a}^{b} f(t)g(t,s) \,dsdt.$$

Consequently

$$\lim\_{r \to \infty} \int\_{-r}^{r} \int\_{a}^{b} f(t)g(t,s)dsdt = \lim\_{r \to \infty} \int\_{a}^{b} \Gamma\_{r}(s)ds = \int\_{a}^{b} \Gamma(s)ds.$$

So by Hake's theorem,

$$\int\_{-\infty}^{\infty} \int\_{a}^{b} f(t)g(t,s) \, dsdt = \int\_{a}^{b} \Gamma(s)ds = \int\_{a}^{b} \int\_{-\infty}^{\infty} f(t)g(t,s)dtds.$$

**Theorem 24.** *Let f* ∈ *BV*0(**R**) *and g* : **R** × **R** → **C** *be a function such that its partial derivative D*2*g is bounded and continuous on* **R** × **R***. If s*<sup>0</sup> ∈ **R** *is such that*


*Then* Γ *is derivable at s*0*, and*

$$
\Gamma'(s\_0) = \int\_{-\infty}^{\infty} f(t) D\_2 g(t, s\_0) dt. \tag{19}
$$

*Proof.* Using conditions (1) and (2) and the Mean Value theorem, there exist *δ >* 0 and *M >* 0 such that, for each *s* ∈ (*s*<sup>0</sup> − *δ*,*s*<sup>0</sup> + *δ*),

$$\left| \int\_{u}^{v} D\_{2} \mathbf{g}(t, s) dt \right| < M \quad \text{and} \quad \left| \int\_{u}^{v} \mathbf{g}(t, s) dt \right| < M,\tag{20}$$

for all [*u*, *v*] ⊆ **R**.

Let *a*, *b* be real numbers with *s*<sup>0</sup> − *δ < a < s*<sup>0</sup> *< b < s*<sup>0</sup> + *δ*. We use Theorem 12 to prove (19). The function *f*(*t*)*g*(*t*, ·) is differentiable on [*a*, *b*] for each *t* ∈ **R**, therefore *f*(*t*)*g*(*t*, ·) is *ACG<sup>δ</sup>* on [*a*, *b*] for all *t* ∈ **R**. By (20) and Theorem 9, *f*(·)*g*(·,*s*) is HK-integrable on **R** for all *s* ∈ [*a*, *b*]. Then

$$\Gamma'(s\_0) = \int\_{-\infty}^{\infty} f(t) D\_2 g(t, s\_0) dt$$

when, if

$$\Gamma\_2 := \int\_{-\infty}^{\infty} f(t) D\_2 g(t\_\prime \cdot) dt$$

is continuous at *s*0, and

$$\int\_{s}^{t} \int\_{-\infty}^{\infty} f(t) D\_{2} \mathbf{g}(t, s) dt ds = \int\_{-\infty}^{\infty} \int\_{s}^{t} f(t) D\_{2} \mathbf{g}(t, s) ds dt$$

for all [*s*, *t*] ⊆ [*a*, *b*]. The first affirmation is true by (20) and Theorem 22, and the second affirmation is true due to (20) and Theorem 23

#### **3.1. Some applications**

An important work about the Fourier transform using the Henstock-Kurzweil integral: existence, continuity, inversion theorems etc. was published in [5]. Nevertheless, there are some omissions in that results that use the Lemma 25 (a) of [5]. Also the authors of this book chapter in [6], [3] and [4] have studied existence, continuity and Riemann-Lebesgue lemma about the Fourier transform of functions belong to *HK*(**R**) ∩ *BV*(**R**) and *BV*0(**R**). Following the line of [6], in Theorem 26 we include some results from them as consequences of theorems above section.

Let *f* and *g* be real-valued functions on **R**. The convolution of *f* and *g* is the function *f* ∗ *g* defined by

$$f \ast g(x) = \int\_{-\infty}^{\infty} f(x - y)g(y) dy$$

for all *x* such that the integral exists. Several conditions can be imposed on *f* and *g* to guarantee that *f* ∗ *g* is defined on **R**. For example, if *f* is HK- integrable and *g* is of bounded variation.

**Lemma 25.** *For f* ∈ *HK*(**R**) ∩ *BV*(**R**)*,* lim |*x*|→∞ *f*(*x*) = 0.

*Proof.* Since *f* is a bounded variation function on **R** then the limit of *f*(*x*), as |*x*| → ∞, exists. Suppose that lim |*x*|→∞ *f*(*x*) = *α* = 0. Take 0 *< <* |*α*|. There exists *A >* 0 such that *α* − *< f*(*x*), for all |*x*| *> A*. Observe that *f*(*x*) *>* 0 on [*A*, ∞), so *f* ∈ *L*([*A*, ∞)). Therefore the constant function *α* − is Lebesgue integrable on [*A*, ∞), which is a contradiction.

Observe, as consequence of above Lemma, we have that the vector space *HK*(**R**) ∩ *BV*(**R**) is contained in *BV*0(**R**). So the next theorem is an immediately consequence of above section.

**Theorem 26.** *If f* ∈ *HK*(**R**) ∩ *BV*(**R**)*, then*

*1. f exists on* **R***.*

16 ime knjige

for all [*u*, *v*] ⊆ **R**.

*s* ∈ [*a*, *b*]. Then

when, if

is continuous at *s*0, and

**3.1. Some applications**

above section.

defined by

 *t s*

 ∞ −∞

affirmation is true due to (20) and Theorem 23

*M >* 0 such that, for each *s* ∈ (*s*<sup>0</sup> − *δ*,*s*<sup>0</sup> + *δ*),

 *v u*

*D*2*g*(*t*,*s*)*dt*

Γ (*s*0) =

Γ<sup>2</sup> :=

*f*(*t*)*D*2*g*(*t*,*s*)*dtds* =

*f* ∗ *g*(*x*) =

 

*Proof.* Using conditions (1) and (2) and the Mean Value theorem, there exist *δ >* 0 and

 *v u*

*f*(*t*)*D*2*g*(*t*,*s*0)*dt*

*f*(*t*)*D*2*g*(*t*, ·)*dt*

 ∞ −∞

for all [*s*, *t*] ⊆ [*a*, *b*]. The first affirmation is true by (20) and Theorem 22, and the second

An important work about the Fourier transform using the Henstock-Kurzweil integral: existence, continuity, inversion theorems etc. was published in [5]. Nevertheless, there are some omissions in that results that use the Lemma 25 (a) of [5]. Also the authors of this book chapter in [6], [3] and [4] have studied existence, continuity and Riemann-Lebesgue lemma about the Fourier transform of functions belong to *HK*(**R**) ∩ *BV*(**R**) and *BV*0(**R**). Following the line of [6], in Theorem 26 we include some results from them as consequences of theorems

Let *f* and *g* be real-valued functions on **R**. The convolution of *f* and *g* is the function *f* ∗ *g*

*f*(*x* − *y*)*g*(*y*)*dy*

 ∞ −∞  *t s*

*f*(*t*)*D*2*g*(*t*,*s*)*dsdt*

*g*(*t*,*s*)*dt*

 

*< M*, (20)

*< M* and

Let *a*, *b* be real numbers with *s*<sup>0</sup> − *δ < a < s*<sup>0</sup> *< b < s*<sup>0</sup> + *δ*. We use Theorem 12 to prove (19). The function *f*(*t*)*g*(*t*, ·) is differentiable on [*a*, *b*] for each *t* ∈ **R**, therefore *f*(*t*)*g*(*t*, ·) is *ACG<sup>δ</sup>* on [*a*, *b*] for all *t* ∈ **R**. By (20) and Theorem 9, *f*(·)*g*(·,*s*) is HK-integrable on **R** for all

> ∞ −∞

 ∞ −∞


$$
\widehat{f}'(s) = -i\widehat{\mathfrak{g}}(s), \text{ for each } s \in \mathbb{R} \text{ (}0\text{)}.
$$

*4. For h* <sup>∈</sup> *<sup>L</sup>*1(**R**) <sup>∩</sup> *BV*(**R**)*,<sup>f</sup>* <sup>∗</sup> *<sup>h</sup>*(*s*) = *<sup>f</sup>* (*s*)*h*(*s*) *for all s* ∈ **R***.*

*Proof.* We observe that

$$\left| \int\_{u}^{v} e^{-its} dt \right| \leq \frac{2}{|s|},\tag{21}$$

for all [*u*, *<sup>v</sup>*] <sup>⊆</sup> **<sup>R</sup>**. Thus, each *<sup>s</sup>*<sup>0</sup> <sup>=</sup> 0 satisfies Hypothesis (**H**) relative to *<sup>e</sup>*−*its*.

(*a*) Theorem 21 implies that *f* (*s*0) exists for all *s*<sup>0</sup> = 0 and, since *f* ∈ *HK*(**R**), *f* (0) exists. Therefore *f* exists on **R**.

(*b*) By Theorem 22, *f* is continuous at *s*0, for all *s*<sup>0</sup> = 0.

(*c*) It follows by Theorem 12 in similar way to the proof of Theorem 24.

(*d*) Let *<sup>k</sup>*(*x*, *<sup>y</sup>*) = *<sup>f</sup>*(*<sup>y</sup>* <sup>−</sup> *<sup>x</sup>*)*e*−*iys*, where *<sup>s</sup>* is a fixed real number. We get, for each *<sup>y</sup>* <sup>∈</sup> **<sup>R</sup>** and all [*u*, *v*] ⊆ **R**,

$$\begin{aligned} \left| \int\_{\mathfrak{u}}^{\upsilon} k(\mathfrak{x}, y) d\mathfrak{x} \right| &= \left| \int\_{\mathfrak{u}}^{\upsilon} f(y - \mathfrak{x}) d\mathfrak{x} \right| \\ &= \left| \int\_{y - \mathfrak{u}}^{y - \upsilon} f(z) dz \right| \le ||f||\_{A}. \end{aligned}$$

So, every real number *y* satisfies Hypothesis (**H**) relative to *k*. Now, observe that *h* ∈ *BV*0(**R**) and *k* is measurable and bounded. Thus, by Theorem 23,

$$\int\_{-a}^{a} \int\_{-\infty}^{\infty} h(\mathbf{x}) k(\mathbf{x}, y) d\mathbf{x} dy = \int\_{-\infty}^{\infty} \int\_{-a}^{a} h(\mathbf{x}) k(\mathbf{x}, y) dy d\mathbf{x},\tag{22}$$

for all *a >* 0.

On the other hand,

$$\begin{aligned} \left| h(\mathbf{x}) \int\_{-a}^{a} f(y-\mathbf{x}) e^{-iys} dy \right| &\leq |h(\mathbf{x})| \left| \int\_{-a-\mathbf{x}}^{a-\mathbf{x}} f(z) e^{-izs} dz \right| \\ &\leq |h(\mathbf{x})| ||f(\cdot) e^{-i(\cdot)s}||\_{A^{\cdot}} \end{aligned}$$

Since *h* ∈ *L*(**R**), using Dominated Convergence theorem, it follows that

$$\begin{aligned} \widehat{f}(s)\widehat{h}(s) &= \int\_{-\infty}^{\infty} h(\mathbf{x}) \int\_{-\infty}^{\infty} f(y-\mathbf{x}) e^{-iys} dy d\mathbf{x} \\ &= \lim\_{a \to \infty} \int\_{-\infty}^{\infty} h(\mathbf{x}) \int\_{-a}^{a} f(y-\mathbf{x}) e^{-iys} dy d\mathbf{x} .\end{aligned}$$

Moreover, from (22), we have

$$\begin{aligned} \widehat{f}(s)\widehat{h}(s) &= \lim\_{a \to \infty} \int\_{-a}^{a} \int\_{-\infty}^{\infty} h(\mathbf{x})f(\mathbf{y}-\mathbf{x})e^{-i y s}d\mathbf{x}d\mathbf{y} \\ &= \lim\_{a \to \infty} \int\_{-a}^{a} (f\*h)(y)e^{-i y s}dy. \end{aligned}$$

We conclude, by Hake's theorem, that

$$
\widehat{f \ast h}(s) = \widehat{f}(s)\widehat{h}(s).
$$

Recall that the Laplace transform, at *z* ∈ **C**, of a function *f* : [0, ∞) → **R** is defined as

$$L(f)(z) = \int\_0^\infty f(t)e^{-zt}dt.$$

**Theorem 27.** *If f* ∈ *HK*([0, ∞)) ∩ *BV*([0, ∞))*, then*


## **4. A set of functions in** *HK*(**R**) <sup>∩</sup> *BV*0(**R**) \ *<sup>L</sup>*1(**R**)

Taking into account Lemma 25, the set *HK*(**R**) ∩ *BV*(**R**) is included in *BV*0(**R**) and does not have inclusion relations with *<sup>L</sup>*1(**R**). Since the step functions belong to *HK*(**R**) <sup>∩</sup> *BV*(**R**), then by Lemma 13, we have that *HK*(**R**) ∩ *BV*(**R**) is dense in *HK*(**R**). In this section we exhibit a set of functions in *HK*(**R**) <sup>∩</sup> *BV*0(**R**) \ *<sup>L</sup>*1(**R**).

**Proposition 28.** *Let b > a >* 0*. Suppose that f* : [*a*, ∞) → **R** *is not identically zero, is continuous and periodic with period b* <sup>−</sup> *a. Let F*(*x*) = *<sup>x</sup> <sup>a</sup> f*(*t*)*dt be bounded on* [*a*, ∞)*. Moreover, assume that ϕ* : [*a*, ∞) → **R** *is a nonnegative and monotone decreasing function which satisfies the next conditions:*


18 ime knjige

for all *a >* 0.

On the other hand,

So, every real number *y* satisfies Hypothesis (**H**) relative to *k*. Now, observe that *h* ∈ *BV*0(**R**)

 ∞ −∞

≤ |*h*(*x*)|

 *a* −*a*

> 

≤ |*h*(*x*)| *f*(·)*e*

*f*(*y* − *x*)*e*

*f*(*y* − *x*)*e*

*h*(*x*)*f*(*y* − *x*)*e*

(*s*)*h*(*s*).

*f*(*t*)*e*

<sup>−</sup>*ztdt*.

<sup>−</sup>*iysdy*.

 *a*−*x* −*a*−*x*

*f*(*z*)*e*

−*i*(·)*s A*.

<sup>−</sup>*iysdydx*

<sup>−</sup>*iysdydx*.

<sup>−</sup>*iysdxdy*

<sup>−</sup>*izsdz* 

*h*(*x*)*k*(*x*, *y*)*dydx*, (22)

*h*(*x*)*k*(*x*, *y*)*dxdy* =

<sup>−</sup>*iysdy* 

> ∞ −∞ *h*(*x*) *a* −*a*

 *a* −*a*

 *a* −*a*

*<sup>f</sup>* <sup>∗</sup> *<sup>h</sup>*(*s*) = *<sup>f</sup>*

Recall that the Laplace transform, at *z* ∈ **C**, of a function *f* : [0, ∞) → **R** is defined as

 ∞ 0

*2. If F*(*x*, *y*) = *L*(*f*)(*x* + *iy*)*, then F*(·, *y*) *is continuous on* **R** *for all y* = 0*, and F*(*x*, ·) *is continuous*

*L*(*f*)(*z*) =

 ∞ −∞

(*f* ∗ *h*)(*y*)*e*

*f*(*y* − *x*)*e*

Since *h* ∈ *L*(**R**), using Dominated Convergence theorem, it follows that

 ∞ −∞ *h*(*x*) ∞ −∞

<sup>=</sup> lim*a*→<sup>∞</sup>

<sup>=</sup> lim*a*→<sup>∞</sup>

(*s*)*<sup>h</sup>*(*s*) = lim*a*→<sup>∞</sup>

and *k* is measurable and bounded. Thus, by Theorem 23,

 ∞ −∞

 *a* −*a*

 *h*(*x*) *a* −*a*

*f*

*f*

**Theorem 27.** *If f* ∈ *HK*([0, ∞)) ∩ *BV*([0, ∞))*, then*

*1. L*(*f*)(*z*) *exists for all z* ∈ **C***.*

*on* **R** *for all x* = 0.

We conclude, by Hake's theorem, that

Moreover, from (22), we have

(*s*)*h*(*s*) =

*Then the product <sup>ϕ</sup> <sup>f</sup>* <sup>∈</sup> *HK*([*a*, <sup>∞</sup>)) \ *<sup>L</sup>*1([*a*, <sup>∞</sup>)).

*Proof.* We take *to* ∈ (*a*, *b*), *δ<sup>o</sup> >* 0 and *γ >* 0 such that

$$
\gamma \le |f(t)| \quad \text{for each} \quad t \in [t\_0 - \delta\_0, t\_0 + \delta\_0] \subset (a, b).
$$

Periodicity of *f* gives

*γ* ≤ | *f*(*t*)|

for each *<sup>t</sup>* <sup>∈</sup> <sup>∞</sup> *<sup>k</sup>*=0[*to* − *δ<sup>o</sup>* + *k*(*b* − *a*), *to* + *δ<sup>o</sup>* + *k*(*b* − *a*)]. Therefore,

$$\begin{split} \int\_{a}^{b+n(b-a)} \varphi(t) |f(t)| dt &\geq \gamma \sum\_{k=0}^{n} \int\_{t\_{0}-\delta\_{0}+k(b-a)}^{t\_{0}+\delta\_{0}+k(b-a)} \varphi(t) dt \\ &\geq \gamma \sum\_{k=0}^{n} \int\_{t\_{0}-\delta\_{0}+k(b-a)}^{t\_{0}+\delta\_{0}+k(b-a)} \varphi(t\_{0}+\delta\_{0}+k(b-a)) dt \\ &= \gamma (2\delta\_{0}) \sum\_{k=0}^{n} \varphi(t\_{0}+\delta\_{0}+k(b-a)). \end{split} \tag{23}$$

Also,

$$\begin{split} \int\_{a}^{b+n(b-a)} \varrho(t)dt &\leq \sum\_{k=0}^{n} \int\_{a+k(b-a)}^{b+k(b-a)} \varrho(t)dt \\ &\leq \sum\_{k=0}^{n} \varrho(a+k(b-a)) \int\_{a+k(b-a)}^{b+k(b-a)} dt \\ &\leq (b-a)\varrho(a) \\ &+ (b-a) \sum\_{k=1}^{n} \varrho(t\_{o} + \delta\_{o} + (k-1)(b-a)). \end{split} \tag{24}$$

Because of *ϕ* ∈/ *HK*([*a*, ∞]), we get lim*n*→<sup>∞</sup> *<sup>b</sup>*+*n*(*b*−*a*) *<sup>a</sup> ϕ*(*t*)*dt* = ∞. Thus, equations (23) and (24) imply *<sup>ϕ</sup> <sup>f</sup>* <sup>∈</sup>/ *<sup>L</sup>*1([*a*, <sup>∞</sup>)]. On the other hand, by Chartier-Dirichlet's Test of [11], the function *ϕ f* belongs to *HK*([*a*, ∞)).

**Corollary 29.** *Let α*, *β be positive numbers such that α* + *β >* 1 *with β* ≤ 1*. Suppose a >* 0 *and <sup>f</sup>* : [*a*, <sup>∞</sup>) <sup>→</sup> **<sup>R</sup>** *obeys the hypotheses of Proposition 28. Then, the function fα*,*<sup>β</sup>* : [*a*1/*α*, <sup>∞</sup>) <sup>→</sup> **<sup>R</sup>** *defined by*

$$f\_{\mathfrak{a},\mathfrak{\beta}}(t) = \frac{f(t^{\mathfrak{a}})}{t^{\mathfrak{\beta}}} \tag{25}$$

*is in HK*([*a*1/*α*, <sup>∞</sup>)) \ *<sup>L</sup>*1([*a*1/*α*, <sup>∞</sup>))*.*

*Proof.* The change of variable *u* = *t <sup>α</sup>* gives,

$$\int\_{a^{\frac{1}{4}}}^{\infty} \frac{f(t^a)}{t^{\beta}} dt = \int\_{a}^{\infty} \frac{f(u)}{u^{\frac{\beta - 1}{a} + 1}} du. \tag{26}$$

The hypotheses for *α*, *β* imply that the function *ϕ*(*u*) = *u*−[ (*β*−1) *<sup>α</sup>* <sup>+</sup>1] satisfies the conditions of Proposition 28. Then, *<sup>ϕ</sup> <sup>f</sup>* <sup>∈</sup> *HK*([*a*, <sup>∞</sup>) \ *<sup>L</sup>*1([*a*, <sup>∞</sup>)), satisfying the statement of the corollary.

**Proposition 30.** *Let β > α >* 0 *be fixed with β* + *α >* 1*. Suppose f* : [*a*, ∞) → **R** *is a bounded and continuous function, with bounded derivative. Then the function fα*,*<sup>β</sup>* : [*a*1/*α*, <sup>∞</sup>) <sup>→</sup> **<sup>R</sup>***, defined by fα*,*β*(*t*) = *f*(*t <sup>α</sup>*)/*tβ, belongs to the space BV*([*a*1/*α*, ∞))*.*

*Proof.* Let *M*<sup>1</sup> and *M*<sup>2</sup> be bounds for *f* and *f* , respectively. We have,

$$f'\_{\alpha,\beta}(t) = \frac{\alpha f'(t^{\alpha})}{t^{\beta-\alpha+1}} - \frac{\beta f(t^{\alpha})}{t^{\beta+1}},$$

which gives

$$\left|f'\_{\alpha,\beta}(t)\right| \le \frac{\alpha M\_2}{t^{\beta-\alpha+1}} + \frac{\beta M\_1}{t^{\beta+1}}.$$

Now, take *x > a* 1 *<sup>α</sup>* . Since *β* − *α >* 0, then

$$\frac{\alpha M\_2}{t^{\beta-\alpha+1}} + \frac{\beta M\_1}{t^{\beta+1}} \in L^1([a^{\frac{1}{\alpha}}, \mathfrak{x})).$$

A straightforward application of the Theorem 7.7 of [11] implies *f <sup>α</sup>*,*<sup>β</sup>* <sup>∈</sup> *<sup>L</sup>*1([*a*1/*α*, *<sup>x</sup>*)). Moreover

$$\begin{split} \int\_{a^{\frac{1}{2}}}^{\infty} |f'\_{\alpha,\beta}(t)|dt &\leq aM\_{2} \int\_{a^{\frac{1}{2}}}^{\infty} t^{-\beta+\alpha-1}dt \\ &+ \beta M\_{1} \int\_{a^{\frac{1}{2}}}^{\infty} t^{-\beta-1}dt \\ &= -\frac{aM\_{2}}{\beta-\alpha} \left( \frac{1}{\mathbf{x}^{\beta-\alpha}} - \frac{1}{a^{\frac{\beta-\alpha}{\alpha}}} \right) \\ &- M\_{1} \left( \frac{1}{\mathbf{x}^{\beta}} - \frac{1}{a^{\frac{\beta}{\alpha}}} \right) \\ &\leq \frac{aM\_{2}}{\beta-\alpha} \frac{1}{a^{\frac{\beta-\alpha}{\alpha}}} + \frac{M\_{1}}{a^{\frac{\beta}{\alpha}}}. \end{split}$$

These estimates together with the Theorem 7.5 of [11] imply,

$$V(f\_{a,\beta};\ \left[a^{\frac{1}{a}},\infty\right)) \le \frac{\alpha M\_2}{\beta - a} \frac{1}{a^{\frac{\beta - a}{a}}} + \frac{M\_1}{a^{\frac{\beta}{a}}}.\tag{27}$$

If *<sup>x</sup>* tends to <sup>∞</sup>, one gets *<sup>f</sup>α*,*<sup>β</sup>* <sup>∈</sup> *BV*([*a*1/*α*, <sup>∞</sup>)).

20 ime knjige

*defined by*

*fα*,*β*(*t*) = *f*(*t*

which gives

Now, take *x > a*

Moreover

1

Because of *ϕ* ∈/ *HK*([*a*, ∞]), we get lim*n*→<sup>∞</sup>

function *ϕ f* belongs to *HK*([*a*, ∞)).

*is in HK*([*a*1/*α*, <sup>∞</sup>)) \ *<sup>L</sup>*1([*a*1/*α*, <sup>∞</sup>))*.*

*Proof.* The change of variable *u* = *t*

*<sup>b</sup>*+*n*(*b*−*a*)

(24) imply *<sup>ϕ</sup> <sup>f</sup>* <sup>∈</sup>/ *<sup>L</sup>*1([*a*, <sup>∞</sup>)]. On the other hand, by Chartier-Dirichlet's Test of [11], the

**Corollary 29.** *Let α*, *β be positive numbers such that α* + *β >* 1 *with β* ≤ 1*. Suppose a >* 0 *and <sup>f</sup>* : [*a*, <sup>∞</sup>) <sup>→</sup> **<sup>R</sup>** *obeys the hypotheses of Proposition 28. Then, the function fα*,*<sup>β</sup>* : [*a*1/*α*, <sup>∞</sup>) <sup>→</sup> **<sup>R</sup>**

*<sup>f</sup>α*,*β*(*t*) = *<sup>f</sup>*(*<sup>t</sup>*

 ∞ *a*

Proposition 28. Then, *<sup>ϕ</sup> <sup>f</sup>* <sup>∈</sup> *HK*([*a*, <sup>∞</sup>) \ *<sup>L</sup>*1([*a*, <sup>∞</sup>)), satisfying the statement of the corollary.

**Proposition 30.** *Let β > α >* 0 *be fixed with β* + *α >* 1*. Suppose f* : [*a*, ∞) → **R** *is a bounded and continuous function, with bounded derivative. Then the function fα*,*<sup>β</sup>* : [*a*1/*α*, <sup>∞</sup>) <sup>→</sup> **<sup>R</sup>***, defined by*

> (*t α*) *<sup>t</sup>β*−*α*+<sup>1</sup> <sup>−</sup> *<sup>β</sup> <sup>f</sup>*(*<sup>t</sup>*

> > *αM*<sup>2</sup> *<sup>t</sup>β*−*α*+<sup>1</sup> <sup>+</sup> *<sup>β</sup>M*<sup>1</sup>

*<sup>t</sup>β*+<sup>1</sup> <sup>∈</sup> *<sup>L</sup>*1([*<sup>a</sup>*

*f*(*u*)

(*β*−1)

, respectively. We have,

*α*) *<sup>t</sup>β*+<sup>1</sup> ,

*<sup>t</sup>β*+<sup>1</sup> .

1 *<sup>α</sup>* , *x*)).

*u β*−1 *<sup>α</sup>* +1

*<sup>α</sup>* gives,

*f*(*t α*) *<sup>t</sup><sup>β</sup> dt* <sup>=</sup>

 ∞ *a* 1 *α*

The hypotheses for *α*, *β* imply that the function *ϕ*(*u*) = *u*−[

*<sup>α</sup>*)/*tβ, belongs to the space BV*([*a*1/*α*, ∞))*.*

*f*

 *f <sup>α</sup>*,*β*(*t*) ≤

*αM*<sup>2</sup> *<sup>t</sup>β*−*α*+<sup>1</sup> <sup>+</sup> *<sup>β</sup>M*<sup>1</sup>

A straightforward application of the Theorem 7.7 of [11] implies *f*

*<sup>α</sup>* . Since *β* − *α >* 0, then

*<sup>α</sup>*,*β*(*t*) = *<sup>α</sup> <sup>f</sup>*

*Proof.* Let *M*<sup>1</sup> and *M*<sup>2</sup> be bounds for *f* and *f*

*α*)

*<sup>a</sup> ϕ*(*t*)*dt* = ∞. Thus, equations (23) and

*<sup>t</sup><sup>β</sup>* (25)

*du*. (26)

*<sup>α</sup>* <sup>+</sup>1] satisfies the conditions of

*<sup>α</sup>*,*<sup>β</sup>* <sup>∈</sup> *<sup>L</sup>*1([*a*1/*α*, *<sup>x</sup>*)).

Corollary 29 and Proposition 30 provide us Henstock-Kurzweil integrable functions defined on unbounded intervals which are not Lebesgue integrable.

**Corollary 31.** *Let a*, *α*, *β be such that:* 0 *< a*, 0 *< α < β* ≤ 1 *and* 1 *< β* + *α. Suppose that f* : [*a*, ∞) → **R** *satisfies both the hypotheses of Corollary 29 and Proposition 30. Then, the function <sup>f</sup>α*,*<sup>β</sup> belongs to HK*([*a*1/*α*, <sup>∞</sup>)) <sup>∩</sup> *BV*([*a*1/*α*, <sup>∞</sup>)) \ *<sup>L</sup>*1([*a*1/*α*, <sup>∞</sup>))*.*

Taking into account the above functions we have the following corollary.

**Corollary 32.** *Let a*, *α*, *β be such that:* 0 *< a*, 0 *< α < β* ≤ 1 *and* 1 *< β* + *α, and let h in BV*([−*a*1/*α*, *<sup>a</sup>*1/*α*])*. Suppose that f* : [*a*, <sup>∞</sup>) <sup>→</sup> **<sup>R</sup>** *satisfies both the hypotheses of Corollary <sup>29</sup> and Proposition 30. Then f* : **R** → **R** *defined by*

$$g(t) = \begin{cases} h(t) & \text{if } t \in (-a^{1/\alpha}, a^{1/\alpha}), \\\\ \frac{f(|t|^{\alpha})}{|t|^{\beta}} & \text{if } f \in (-\infty, -a^{1/\alpha}] \cup [a^{1/\alpha}, \infty). \end{cases}$$

*is in HK*(**R**) ∩ *BV*(**R**) \ *L*(**R**).

**Example 33.** *Let us consider the trigonometric functions* sin(*t*) *and* cos(*t*)*. Then the following family of functions satisfies the hypotheses of Theorem 34.*

$$\begin{aligned} \sin^{\alpha}\_{\beta} : \mathbb{R} \to \mathbb{R}; \quad \sin^{\alpha}\_{\beta}(t) = \chi\_{1\kappa}(t) \frac{\sin(t^{\alpha})}{t^{\beta}}, \\\cos^{\alpha}\_{\beta} : \mathbb{R} \to \mathbb{R}; \quad \cos^{\alpha}\_{\beta}(t) = \chi\_{2\kappa}(t) \frac{\cos(t^{\alpha})}{t^{\beta}}. \end{aligned}$$

$$\mathbf{0}$$

*Here χ*1,*<sup>α</sup> and χ*2,*<sup>α</sup> are the characteristic functions of the intervals* [*π*1/*α*, ∞) *and* [(*π*/2)1/*α*, ∞)*, respectively. The numbers α*, *β are taken as in Corollary 31.*

From the above example belongs to *HK*(**R**) ∩ *BV*(**R**) \ *L*( **R**). By the Multiplier theorem it follows that *HK*(**R**) <sup>∩</sup> *BV*(**R**) <sup>⊂</sup> *<sup>L</sup>*2(**R**), so the above function is in *BV*0(**R**) <sup>∩</sup> *<sup>L</sup>*2(**R**) \ *<sup>L</sup>*( **<sup>R</sup>**). Therefore, there exist functions in *<sup>L</sup>*2(**R**) \ *<sup>L</sup>*( **<sup>R</sup>**) such that their Fourier transforms exist as in (1), as an integral in HK sense.

## **5. The Riemann-Lebesgue Lemma and the Dirichlet-Jordan Theorem for** *BV*<sup>0</sup> **functions**

The Riemann-Lebesgue lemma is a fundamental result of the Harmonic Analysis. An novel aspect is the validity of this lemma for functions which are not Lebesgue integrable, since this fact could help to expand the space of functions where the inversion of the Fourier transform is possible. In this section we prove a generalization of the Riemann-Lebesgue Lemma for functions of bounded variation which vanish at infinity. As consequence, it is obtained a proof of the Dirichlet-Jordan theorem for this kind of functions. This theorem provides a pointwise inversion of the Fourier transform.

We observe that the implications 1 and 2 of Theorem 26 are particularizations of the next result.

**Theorem 34** (Generalization of Riemann-Lebesgue Lemma)**.** *Let ϕ* ∈ *HKloc*(**R**) *be a function such that* Φ(*t*) = *<sup>t</sup>* <sup>0</sup> *ϕ*(*x*)*dx is bounded function on* **R***. If f* ∈ *BV*0(**R**), *then the function H*(*w*) = <sup>∞</sup> <sup>−</sup><sup>∞</sup> *<sup>f</sup>*(*t*)*ϕ*(*wt*)*dt is defined on* **<sup>R</sup>** \ {0} , *it is continuous and*

$$\lim\_{|w| \to \infty} H(w) = 0.$$

*Proof.* Given *w* ∈ **R**, we define *ϕw*(*t*) = *ϕ*(*wt*). Because of *ϕ* ∈ *HKloc*(**R**), then *ϕ* and *ϕ<sup>w</sup>* are in *HK*([0, *b*]), for *b >* 0. By Jordan decomposition, there exist functions *f*<sup>1</sup> and *f*<sup>2</sup> which are nondecreasing functions belonging to *BV*0(**R**) such that *f* = *f*<sup>1</sup> − *f*2. Hence, by Chartier-Dirichlet's Test, *f ϕ<sup>w</sup>* ∈ *HK*([0, ∞]). By applying the Multiplier Theorem and supposing *w* = 0, it follows

$$\begin{split} \int\_{0}^{\infty} f(t)\varrho(wt)dt &= -\int\_{0}^{\infty} \frac{\Phi(wt)}{w} df(t) \\ &= -\int\_{0}^{\infty} \frac{\Phi(wt)}{w} df\_{1}(t) \\ &+ \int\_{0}^{\infty} \frac{\Phi(wt)}{w} df\_{2}(t). \end{split} \tag{28}$$

where *d fi*(*t*) is the Lebesgue-Sieltjes measure generated by *fi*, *i* = 1, 2. Let *β* a positive number and let *M* the upper bound of |Φ|. For *w* ∈ [*β*, ∞) we have that

$$\left|\frac{\Phi(wt)}{w}\right| \le \frac{M}{\beta}.\tag{29}$$

Since Φ(*wt*)/*w* is continuous over [*β*, ∞) and the measures *d fi*(*t*) are finite, then by the Dominated Convergence Theorem applied to right side integrals in (28), it follows that

$$\lim\_{w \to w\_0} H(w) = H(w\_0)\_\*$$

for each *w*<sup>0</sup> ∈ [*β*, ∞). Since *β* is arbitrary, we obtain the continuity of *H* on (0, ∞). Moreover, by (28), we have for *w* ∈ (0, ∞) that

$$\left| \int\_0^\infty f(t)\varrho(wt)dt \right| \le \frac{M}{|w|}Var(f;[0,\infty]).$$

Thus, we conclude that

22 ime knjige

*Here χ*1,*<sup>α</sup> and χ*2,*<sup>α</sup> are the characteristic functions of the intervals* [*π*1/*α*, ∞) *and* [(*π*/2)1/*α*, ∞)*,*

From the above example belongs to *HK*(**R**) ∩ *BV*(**R**) \ *L*( **R**). By the Multiplier theorem it follows that *HK*(**R**) <sup>∩</sup> *BV*(**R**) <sup>⊂</sup> *<sup>L</sup>*2(**R**), so the above function is in *BV*0(**R**) <sup>∩</sup> *<sup>L</sup>*2(**R**) \ *<sup>L</sup>*( **<sup>R</sup>**). Therefore, there exist functions in *<sup>L</sup>*2(**R**) \ *<sup>L</sup>*( **<sup>R</sup>**) such that their Fourier transforms exist as

**5. The Riemann-Lebesgue Lemma and the Dirichlet-Jordan Theorem for**

The Riemann-Lebesgue lemma is a fundamental result of the Harmonic Analysis. An novel aspect is the validity of this lemma for functions which are not Lebesgue integrable, since this fact could help to expand the space of functions where the inversion of the Fourier transform is possible. In this section we prove a generalization of the Riemann-Lebesgue Lemma for functions of bounded variation which vanish at infinity. As consequence, it is obtained a proof of the Dirichlet-Jordan theorem for this kind of functions. This theorem provides a

We observe that the implications 1 and 2 of Theorem 26 are particularizations of the next

**Theorem 34** (Generalization of Riemann-Lebesgue Lemma)**.** *Let ϕ* ∈ *HKloc*(**R**) *be a function*

*Proof.* Given *w* ∈ **R**, we define *ϕw*(*t*) = *ϕ*(*wt*). Because of *ϕ* ∈ *HKloc*(**R**), then *ϕ* and *ϕ<sup>w</sup>* are in *HK*([0, *b*]), for *b >* 0. By Jordan decomposition, there exist functions *f*<sup>1</sup> and *f*<sup>2</sup> which are nondecreasing functions belonging to *BV*0(**R**) such that *f* = *f*<sup>1</sup> − *f*2. Hence, by Chartier-Dirichlet's Test, *f ϕ<sup>w</sup>* ∈ *HK*([0, ∞]). By applying the Multiplier Theorem and

= −

Let *β* a positive number and let *M* the upper bound of |Φ|. For *w* ∈ [*β*, ∞) we have that

+ ∞ 0

*H*(*w*) = 0.

 ∞ 0

 ∞ 0

Φ(*wt*) *<sup>w</sup> d f*(*t*)

Φ(*wt*)

Φ(*wt*)

*<sup>w</sup> d f*2(*t*),

*<sup>w</sup> d f*1(*t*) (28)

lim |*w*|→∞

*f*(*t*)*ϕ*(*wt*)*dt* = −

where *d fi*(*t*) is the Lebesgue-Sieltjes measure generated by *fi*, *i* = 1, 2.

<sup>0</sup> *ϕ*(*x*)*dx is bounded function on* **R***. If f* ∈ *BV*0(**R**), *then the function H*(*w*) =

*respectively. The numbers α*, *β are taken as in Corollary 31.*

in (1), as an integral in HK sense.

pointwise inversion of the Fourier transform.

<sup>−</sup><sup>∞</sup> *<sup>f</sup>*(*t*)*ϕ*(*wt*)*dt is defined on* **<sup>R</sup>** \ {0} , *it is continuous and*

 ∞ 0

*BV*<sup>0</sup> **functions**

result.

<sup>∞</sup>

*such that* Φ(*t*) = *<sup>t</sup>*

supposing *w* = 0, it follows

$$\lim\_{|w| \to \infty} \int\_0^\infty f(t)\,\varrho(wt)dt = 0.$$

To complete the proof, we use similar arguments for the interval (−∞, 0].

The above theorem confirms that *H* ∈ *C*0(**R** \ {0}), for each *f* ∈ *BV*0(**R**). As corollary we have the Riemann-Lebesgue Lemma.

**Corollary 35.** *If f* <sup>∈</sup> *BV*0(**R**), *then* <sup>ˆ</sup> *f* ∈ *C*0(**R** \ {0})*.*

We know that if *g*, *h* ∈ *BV*([*a*, ∞]) then *gh* ∈ *BV*([*a*, ∞]). Employing this fact and Theorem 34 we get the following corollary.

**Corollary 36.** *Suppose that δ*, *α >* 0 *and f* ∈ *BV*(**R**), *then*

$$\lim\_{M \to \infty} \int\_{\delta}^{\infty} \frac{f(t)}{t^{\alpha}} e^{-iMt} dt = 0.$$

The Sine Integral is defined as

$$Si(x) = \frac{2}{\pi} \int\_0^x \frac{\sin t}{t} dt\_\prime$$

which has the properties:


We use the Sine Integral function in the proof of the following lemma.

**Lemma 37.** *Let δ >* 0. *If f* ∈ *BV*0(**R**), *then*

$$\lim\_{\varepsilon \to 0} \int\_{\delta}^{\infty} f(t) \frac{\sin \varepsilon t}{t} dt = 0.$$

*Proof.* By Lemma 10 we have

$$\left| \int\_{\delta}^{\infty} \frac{\sin \varepsilon t}{t} f(t) dt \right| \le \left| \int\_{\delta}^{\infty} \left( \int\_{\delta \varepsilon}^{t \varepsilon} \frac{\sin u}{u} du \right) df(t) \right|. \tag{30}$$

Since for each *t* ∈ [*a*, ∞): lim*ε*→<sup>0</sup> *tε δε* sin *u <sup>u</sup> du* = 0 and *tε δε* sin *u <sup>u</sup> du* <sup>≤</sup> *<sup>π</sup>Si*(*π*) for all *<sup>ε</sup> <sup>&</sup>gt;* 0. Then, we obtain the result applying the Lebesgue Dominated Convergence theorem to the integral on the right in (30).

**Lemma 38.** *Suppose that* 0 *< α < β or α < β <* 0. *If f* ∈ *BV*0(**R**), *then for all s* ∈ [*α*, *β*] *we have*

$$\lim\_{\substack{a \to -\infty \\ b \to \infty}} \int\_{a}^{\emptyset} e^{i \text{xs}} \int\_{a}^{b} f(t) e^{-i \text{st}} dt ds = \int\_{a}^{\emptyset} e^{i \text{xs}} \int\_{-\infty}^{\infty} f(t) e^{-i \text{st}} dt ds. \tag{31}$$

*Proof.* We will do the proof for 0 *< α < β*. Let *f* <sup>0</sup>*b*(*s*) = *<sup>b</sup>* <sup>0</sup> *<sup>f</sup>*(*t*)*e*−*istdt* and *<sup>f</sup>* <sup>0</sup>(*s*) = <sup>∞</sup> <sup>0</sup> *<sup>f</sup>*(*t*)*e*−*istdt*, which are continuous on **<sup>R</sup>** {0}. Therefore the integrals in (31) exist. We know that there is *R >* 0 such that | *f*(*t*)| ≤ *R* for all *t* ∈ **R**, and that for any *b >* 0 : *V*(*f* ; [0, *b*]) ≤ *V*(*f* ; [0, ∞)). For each *s* ∈ [*α*, *β*] the Multiplier theorem implies

$$\left| \widehat{f}\_{0b}(\mathbf{s}) \right| \leq \frac{2}{\mathfrak{a}} \left\{ R + V(f; [0, \infty)) \right\} = N.$$

This inequality implies that for any *b >* 0 and all *s* ∈ [*α*, *β*] : *eixs f* <sup>0</sup>*b*(*s*) <sup>≤</sup> *<sup>N</sup>*, for each *<sup>x</sup>* <sup>∈</sup> **<sup>R</sup>**. Applying the theorem of Hake we have: lim*b*→<sup>∞</sup> *<sup>f</sup>* <sup>0</sup>*b*(*s*) = *f* <sup>0</sup>(*s*). Then, by the Lebesgue Dominated Convergence theorem,

$$\lim\_{b \to \infty} \int\_{\alpha}^{\beta} e^{i\mathbf{x}s} \widehat{f}\_{0b}(s) ds = \int\_{\alpha}^{\beta} e^{i\mathbf{x}s} \widehat{f}\_{0}(s) ds.$$

To conclude the proof, we follow a similar process over the interval [*a*, 0] leading *a* to minus infinity.

To obtain the Dirichlet-Jordan theorem we state the following lemma [22, Theorem 11.8].

**Lemma 39.** *Let δ >* 0*. If g is of bounded variation on* [0, *δ*], *then*

24 ime knjige

<sup>∞</sup>

infinity.

We use the Sine Integral function in the proof of the following lemma.

lim *ε*→0

*<sup>t</sup> <sup>f</sup>*(*t*)*dt*

 *tε δε* sin *u*

*f*(*t*)*e*

*V*(*f* ; [0, *b*]) ≤ *V*(*f* ; [0, ∞)). For each *s* ∈ [*α*, *β*] the Multiplier theorem implies

 ∞ *δ*

> ≤ ∞ *δ*

*f*(*t*)

sin *εt t*

> *tε δε*

*<sup>u</sup> du* = 0 and

Then, we obtain the result applying the Lebesgue Dominated Convergence theorem to the

**Lemma 38.** *Suppose that* 0 *< α < β or α < β <* 0. *If f* ∈ *BV*0(**R**), *then for all s* ∈ [*α*, *β*] *we have*

 *β α e ixs* ∞ −∞

*<sup>α</sup>* {*<sup>R</sup>* <sup>+</sup> *<sup>V</sup>*(*<sup>f</sup>* ; [0, <sup>∞</sup>))} <sup>=</sup> *<sup>N</sup>*.

 *β α e ixs f* <sup>0</sup>(*s*)*ds*.

To conclude the proof, we follow a similar process over the interval [*a*, 0] leading *a* to minus

To obtain the Dirichlet-Jordan theorem we state the following lemma [22, Theorem 11.8].

*f*(*t*)*e*

<sup>0</sup>*b*(*s*) = *<sup>b</sup>*

 *eixs f* <sup>0</sup>*b*(*s*) 

<sup>0</sup>*b*(*s*) = *f*

<sup>−</sup>*istdtds* =

<sup>0</sup> *<sup>f</sup>*(*t*)*e*−*istdt*, which are continuous on **<sup>R</sup>** {0}. Therefore the integrals in (31) exist. We know that there is *R >* 0 such that | *f*(*t*)| ≤ *R* for all *t* ∈ **R**, and that for any *b >* 0 :

*dt* = 0.

sin *u u du d f*(*t*) 

> *tε δε* sin *u <sup>u</sup> du*

. (30)

<sup>≤</sup> *<sup>π</sup>Si*(*π*) for all *<sup>ε</sup> <sup>&</sup>gt;* 0.

<sup>−</sup>*istdtds*. (31)

 <sup>0</sup>(*s*) =

<sup>≤</sup> *<sup>N</sup>*, for each *<sup>x</sup>* <sup>∈</sup> **<sup>R</sup>**.

<sup>0</sup>(*s*). Then, by the Lebesgue

<sup>0</sup> *<sup>f</sup>*(*t*)*e*−*istdt* and *<sup>f</sup>*

**Lemma 37.** *Let δ >* 0. *If f* ∈ *BV*0(**R**), *then*

 ∞ *δ*

sin *εt*

*Proof.* By Lemma 10 we have

Since for each *t* ∈ [*a*, ∞): lim*ε*→<sup>0</sup>

lim *<sup>a</sup>*→−<sup>∞</sup> *b*→∞

 *β α e ixs b a*

*Proof.* We will do the proof for 0 *< α < β*. Let *f*

 *f* <sup>0</sup>*b*(*s*) ≤ 2

This inequality implies that for any *b >* 0 and all *s* ∈ [*α*, *β*] :

lim *b*→∞  *β α e ixs f* <sup>0</sup>*b*(*s*)*ds* =

Applying the theorem of Hake we have: lim*b*→<sup>∞</sup> *<sup>f</sup>*

Dominated Convergence theorem,

integral on the right in (30).

$$\lim\_{M \to \infty} \frac{2}{\pi} \int\_0^\delta g(t) \frac{\sin Mt}{t} dt = g(0+)$$

**Theorem 40** (Dirichlet-Jordan Theorem)**.** *If f is a function in BV*0(**R**), *then, for each x* ∈ **R**,

$$\lim\_{\substack{M\to\infty\\\varepsilon\to0}}\frac{1}{2\pi}\int\_{\varepsilon<|s|$$

*Proof.* Let *g*(*x*, *t*) = *f*(*x* − *t*) + *f*(*x* + *t*) and suppose that *δ >* 0. By Fubini's theorem for the Lebesgue integral [22, Theorem 15.7] at [−*M*, −*ε*] × [*a*, *b*] and [*ε*, *M*] × [*a*, *b*] and Lema 38, we have

$$\begin{split} \int\_{\varepsilon<|s|$$

In [*δ*, ∞], by Corollary 36 and Lemma 37, we get

$$\lim\_{M \to \infty, \varepsilon \to 0} \int\_{\delta}^{\infty} \frac{g(\mathbf{x}, t)}{t} (\sin M \, t - \sin \varepsilon t) dt = 0. \tag{33}$$

In [0, *δ*], applying the Lebesgue Dominate Convergence theorem,

$$\lim\_{\varepsilon \to 0} \int\_0^\delta \frac{\mathbf{g}(\mathbf{x}, t)}{t} \sin \varepsilon t dt = 0. \tag{34}$$

Now, by Lemma 39,

$$\lim\_{M \to \infty} \int\_0^\delta g(\mathbf{x}, t) \frac{\sin Mt}{t} dt = g(\mathbf{x}, 0+) = \frac{\pi}{2} \left[ f(\mathbf{x} - \mathbf{0}) + f(\mathbf{x} + \mathbf{0}) \right].$$

We conclude the proof combining (33), (34) and the above expression.

We observe that the classical theorem of Dirichlet-Jordan on *L*(**R**) is a particular case of Theorem 40. Taking into account that *HK*(**R**) ∩ *BV*(**R**) ⊂ *BV*0(**R**), then from Theorem 34 and Theorem 40 we get that if *f* ∈ *HK*(**R**) ∩ *BV*(**R**), then its Fourier transform *f* (*s*) exists in each *s* ∈ **R**; *f* ∈ *C*0(**R**\ {0}), and the expression (32) holds for each *x* ∈ **R**.

**Corollary 41.** *There exist functions in L*2(**R**) \ *<sup>L</sup>*( **<sup>R</sup>**) *such that their Fourier transforms exist as in* (1) *and, for each x* ∈ **R***, the expression* (32) *is true.*

## **6. Conclusions**

We present theorems about convergence of integrals of products in the completion of *HK*(*I*), which those we have a version of Riemann-Lebesgue Lemma (over compact intervals) and analogous results at Riemann-Lebesgue property, a characterization of behavior of *n*-th partial sum of the Fourier series. Moreover, we have gotten basic properties (existence as integral, continuity, asymptotic behavior) about Fourier transform using Henstock-Kurzweil Integral, for this was necessary to get a generalization of Riemann-Lebesgue Lemma over *BV*0(**R**), in particular those characteristics are valid over *HK*(**R**) ∩ *BV*(**R**). This intersection does not have relation inclusion with Lebesgue integrable functions space, we give a set of functions such that it belongs to *HK*(**R**) ∩ *BV*(**R**) \ *L*(**R**). Finally we have a generalization of Dirichlet-Jordan over *BV*0(**R**).

## **Acknowledgments**

The authors acknowledge the support provided by the VIEP-BUAP, Puebla, Mexico.

## **Author details**

Francisco J. Mendoza-Torres1, Ma. Guadalupe Morales-Macías2, Salvador Sánchez-Perales3, and Juan Alberto Escamilla-Reyna1

1 Facultad de Ciencias Físico-Matemáticas, Benemérita Universidad Autónoma de Puebla, Puebla, Mexico

2 Departamento de Matemáticas, Universidad Autónoma Metropolitana-Iztapalapa, México, D. F., Mexico

3 Instituto de Física y Matemáticas, Universidad Tecnológica de la Mixteca, Oaxaca, Mexico

### **References**

26 ime knjige

Now, by Lemma 39,

each *s* ∈ **R**; *f*

**6. Conclusions**

Dirichlet-Jordan over *BV*0(**R**).

and Juan Alberto Escamilla-Reyna1

**Acknowledgments**

**Author details**

Puebla, Mexico

Mexico

México, D. F., Mexico

lim *M*→∞  *δ* 0

(1) *and, for each x* ∈ **R***, the expression* (32) *is true.*

*g*(*x*, *t*)

sin *Mt t*

We conclude the proof combining (33), (34) and the above expression.

*dt* <sup>=</sup> *<sup>g</sup>*(*x*, 0+) = *<sup>π</sup>*

We observe that the classical theorem of Dirichlet-Jordan on *L*(**R**) is a particular case of Theorem 40. Taking into account that *HK*(**R**) ∩ *BV*(**R**) ⊂ *BV*0(**R**), then from Theorem 34

**Corollary 41.** *There exist functions in L*2(**R**) \ *<sup>L</sup>*( **<sup>R</sup>**) *such that their Fourier transforms exist as in*

We present theorems about convergence of integrals of products in the completion of *HK*(*I*), which those we have a version of Riemann-Lebesgue Lemma (over compact intervals) and analogous results at Riemann-Lebesgue property, a characterization of behavior of *n*-th partial sum of the Fourier series. Moreover, we have gotten basic properties (existence as integral, continuity, asymptotic behavior) about Fourier transform using Henstock-Kurzweil Integral, for this was necessary to get a generalization of Riemann-Lebesgue Lemma over *BV*0(**R**), in particular those characteristics are valid over *HK*(**R**) ∩ *BV*(**R**). This intersection does not have relation inclusion with Lebesgue integrable functions space, we give a set of functions such that it belongs to *HK*(**R**) ∩ *BV*(**R**) \ *L*(**R**). Finally we have a generalization of

The authors acknowledge the support provided by the VIEP-BUAP, Puebla, Mexico.

Francisco J. Mendoza-Torres1, Ma. Guadalupe Morales-Macías2, Salvador Sánchez-Perales3,

1 Facultad de Ciencias Físico-Matemáticas, Benemérita Universidad Autónoma de Puebla,

2 Departamento de Matemáticas, Universidad Autónoma Metropolitana-Iztapalapa,

3 Instituto de Física y Matemáticas, Universidad Tecnológica de la Mixteca, Oaxaca,

and Theorem 40 we get that if *f* ∈ *HK*(**R**) ∩ *BV*(**R**), then its Fourier transform *f*

∈ *C*0(**R**\ {0}), and the expression (32) holds for each *x* ∈ **R**.

<sup>2</sup> [ *<sup>f</sup>*(*<sup>x</sup>* <sup>−</sup> <sup>0</sup>) + *<sup>f</sup>*(*<sup>x</sup>* <sup>+</sup> <sup>0</sup>)] .

(*s*) exists in


**Provisional chapter**

## **Double Infinitesimal Fourier Transform Double Infinitesimal Fourier Transform**

## Takashi Gyoshin Nitta

Takashi Gyoshin Nitta

28 ime knjige

[16] Heuser H. Functional Analysis. New York: John Wiley and Sons; 1982.

http://www.jstor.org/stable/2307003?seq=1

http://arxiv.org/pdf/math/0101012.pdf

907-918. Available from: http://tinyurl.com/ke4gfkh

Singapore: World Scientific; 1989.

106 Fourier Transform - Signal Processing and Physical Sciences

Company Reading; 1974.

[17] Riesz M, Livingston AE. A Short Proof of a Classical Theorem in the Theory of Fourier Integrals. Amer. Math. Monthly 62(6) Jun. - Jul. 1955, 434-437. Available from:

[18] Talvila E. Necessary and sufficient conditions for differentiating under the integral sign. Amer. Math. Monthly 2001(108): 544-548. Available from:

[19] Peng-Yee L. Lanzhou Lectures on Henstock Integration. Series in Real Analysis Vol.2.

[21] Talvila E. Limits and Henstock integrals of products. Real Anal. Exch. 1999/00;25(2):

[22] Apostol T. Mathematical Analysis. Massachusetts: Addison-Wesley Publishing

[20] Swartz Ch. Introduction to Gauge Integrals. Singapore: World Scientific; 2001.

Additional information is available at the end of the chapter

Additional information is available at the end of the chapter

http://dx.doi.org/10.5772/59963

## **1. Introduction**

10.5772/59963

For Fourier transform theory, one of the most important and difficult things is how to treat the Dirac delta function and how to define it. In 1930, the Dirac delta function was defined originally by Paul A.M. Dirac([1]) in order to create the quantum mechanical theory in Physics. In classical mechanics, there is the beautiful Newton's law. Under it, it is assumed that a particle is a point with a mass. For the investigation of a small world for example, elementary particles, it should be changed to the quantum mechanical theory where particles are not already only points as in Mathematics but also some area with infinitesimal length for us. They have some properties like waves. The Dirac delta function is defined to be realized the image of the particle in the small world. The particle changed to be the moving wave, and it becomes a set of such waves. It is called field in Physics and we need the second quantization . The quantum mechanics is developed to the quantum field theory where the delta function is much complicated to treat in the standard mathematical theory.

The delta function is usually defined as the delta measure in the functional analysis. Under the basic definition, the functional analysis is developed in the functional space for example Banach space, Hilbert space. These theory is applied to the existence problem of solutions for the ordinary and partial differential equations. On the other hand, the delta function is also defined just as a function in the extension of the real number field ([3],[4],[5],[18]) in 1962. The idea is that firstly the real and complex number fields are extended to the nonstandard universe, secondly the delta function is defined as a function in the extended universe (cf. [3]).

In this chapter the real number field and complex number field are extended twice and a higher degree of delta function is defined as a function on the space of functions. By using the secondly extended delta function, the Fourier transform theory is considered, that is called " double infinitesimal Fourier transform ". In the theory, the Poisson summation formula is also satisfied, and some important examples are calculated. The Fourier transforms of *δ*, *δ*2, ... , and √*δ*, ... can be calculated, which are constant functions as 1, infinite, ... , and infinitesimal, ... .

the terms of the Creative Commons Attribution License (http://creativecommons.org/licenses/by/3.0), which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited. © 2015 The Author(s). Licensee InTech. This chapter is distributed under the terms of the Creative Commons Attribution License (http://creativecommons.org/licenses/by/3.0), which permits unrestricted use, distribution, and eproduction in any medium, provided the original work is properly cited.

©2012 prezimena autora, kod vise prvi et al., licensee InTech. This is an open access chapter distributed under

Then the Fourier transform of the gaussian functional is also calculated. The gaussian functional means that the standard part of the image for *<sup>α</sup>* <sup>∈</sup> *<sup>L</sup>*<sup>2</sup> is exp <sup>−</sup>*πξ* <sup>∞</sup> <sup>−</sup><sup>∞</sup> *<sup>α</sup>*2(*t*)*dt* , for *<sup>ξ</sup>* <sup>∈</sup> **<sup>C</sup>** with Re(*ξ*) <sup>&</sup>gt; 0. The double infinitesimal Fourier transform is calculated as *<sup>C</sup><sup>ξ</sup>* exp −*πξ*−<sup>1</sup> <sup>∞</sup> <sup>−</sup><sup>∞</sup> *<sup>α</sup>*2(*t*)*dt* for *<sup>α</sup>* <sup>∈</sup> *<sup>L</sup>*2(**R**), in which *<sup>C</sup><sup>ξ</sup>* is a constant independent of *<sup>α</sup>* .

Finally a sort of functional is constructed in the theory that associates to Riemann's zeta function. The path integral is defined for the application in the theory, and it is shown that the path integral of the functional *Zs* corresponds to Riemann's zeta function in the case that Re(*s*) > 1. By using the Poisson summation formula for the functional, a relationship appears between the functional and Riemann's zeta function.

## **2. Infinitesimal Fourier transform**

The usual universe is extended, in order to treat many stages of delta functions and functions on the space of functions. For the extension, there exists two methods, one is the second extension of the universe in the nonstandard analysis ([8],[9]) and the other is the Relative set theory in the axiomatic set theory ( [13]). The first one ([8],[9]) is explained here, by using an ultrafilter .

#### **2.1. First extension of the universe**

Let <sup>Λ</sup> be an infinite set. Let *<sup>F</sup>* be a nonprincipal ultrafilter on <sup>Λ</sup>. For each *<sup>λ</sup>* <sup>∈</sup> <sup>Λ</sup>, let *<sup>S</sup><sup>λ</sup>* be a set. An equivalence relation <sup>∼</sup> is induced from *<sup>F</sup>* on <sup>∏</sup>*λ*∈<sup>Λ</sup> *<sup>S</sup>λ*. For *<sup>α</sup>* = (*αλ*), *<sup>β</sup>* = (*βλ*) (*<sup>λ</sup>* <sup>∈</sup> Λ),

$$
\mathfrak{a} \sim \mathfrak{F} \Longleftrightarrow \{ \lambda \in \Lambda \, | \, \mathfrak{a}\_{\lambda} = \mathfrak{f}\_{\lambda} \} \in F. \tag{1}
$$

The set of equivalence classes is called *ultraproduct* of *<sup>S</sup><sup>λ</sup>* for *<sup>F</sup>* with respect to <sup>∼</sup>. If *<sup>S</sup><sup>λ</sup>* <sup>=</sup> *<sup>S</sup>* for *<sup>λ</sup>* <sup>∈</sup> <sup>Λ</sup>, then it is called *ultraproduct* of *<sup>S</sup>* for *<sup>F</sup>* and it is written as <sup>∗</sup>*S*. The set *<sup>S</sup>* is naturally embedded in ∗*S* by the following mapping :

$$s \left( \in S \right) \mapsto \left[ (s\_{\lambda} = s), \lambda \in \Lambda \right] \left( \in \prescript{\*}{}{\mathcal{S}} \right) \tag{2}$$

where [ ] denotes the equivalence class with respect to the ultrafilter *F*. The mapping is written as <sup>∗</sup>, and call it naturally elementary embedding. From now on, we identify the image <sup>∗</sup>(*S*) as *<sup>S</sup>*.

#### **Definition 2.1.1.**

Let *<sup>H</sup>* (<sup>∈</sup> <sup>∗</sup>**Z**) be an infinite even number. The infinite number *<sup>H</sup>* is even, when for *<sup>H</sup>* <sup>=</sup> [(*Hλ*), *<sup>λ</sup>* <sup>∈</sup> <sup>Λ</sup>], {*<sup>λ</sup>* <sup>∈</sup> <sup>Λ</sup> <sup>|</sup> *<sup>H</sup><sup>λ</sup>* is even} ∈ *<sup>F</sup>*. The number <sup>1</sup> *<sup>H</sup>* is written as *<sup>ε</sup>*. We define an infinitesimal lattice space **L**, an infinitesimal lattice subspace *L* and a space of functions *R*(*L*) on *L* as follows :

$$\begin{aligned} \mathbf{L} &:= \varepsilon \, ^\*\mathbf{Z} = \{ \varepsilon z \, | \, z \in \, ^\*\mathbf{Z} \}, \\ L &:= \left\{ \varepsilon z \, \middle| \, z \in \, ^\*\mathbf{Z} \, | \, -\frac{H}{2} \le \varepsilon z < \frac{H}{2} \right\} (\subset \mathbf{L}). \end{aligned}$$

*<sup>R</sup>*(*L*) :<sup>=</sup> {*<sup>ϕ</sup>* <sup>|</sup> *<sup>ϕ</sup>* is an internal function from *<sup>L</sup>* to <sup>∗</sup>**C**} .

The space *R*(*L*) is extended to the space of periodic functions on **L** with period *H*. We write the same notation *R*(*L*) for the space of periodic functions.

Gaishi Takeuchi([18]) introduced an infinitesimal *δ* function. Furthermore Moto-o Kinoshita ([4],[5]) constructed an infinitesimal Fourier transformation theory on *R*(*L*). It is explained briefly.

#### **Definition 2.1.2.**

2 ime knjige

*Cξ* exp 

an ultrafilter .

image <sup>∗</sup>(*S*) as *<sup>S</sup>*. **Definition 2.1.1.**

on *L* as follows :

*L* := *εz* 

**<sup>L</sup>** :<sup>=</sup> *<sup>ε</sup>* <sup>∗</sup>**<sup>Z</sup>** <sup>=</sup> {*ε<sup>z</sup>* <sup>|</sup> *<sup>z</sup>* <sup>∈</sup> <sup>∗</sup>**Z**},

*<sup>z</sup>* <sup>∈</sup> <sup>∗</sup>**Z**, <sup>−</sup> *<sup>H</sup>*

Λ),

−*πξ*−<sup>1</sup> <sup>∞</sup>

<sup>−</sup><sup>∞</sup> *<sup>α</sup>*2(*t*)*dt*

**2. Infinitesimal Fourier transform**

**2.1. First extension of the universe**

embedded in ∗*S* by the following mapping :

[(*Hλ*), *<sup>λ</sup>* <sup>∈</sup> <sup>Λ</sup>], {*<sup>λ</sup>* <sup>∈</sup> <sup>Λ</sup> <sup>|</sup> *<sup>H</sup><sup>λ</sup>* is even} ∈ *<sup>F</sup>*. The number <sup>1</sup>

2 

(<sup>⊂</sup> **<sup>L</sup>**),

<sup>2</sup> <sup>≤</sup> *<sup>ε</sup><sup>z</sup>* <sup>&</sup>lt; *<sup>H</sup>*

appears between the functional and Riemann's zeta function.

Then the Fourier transform of the gaussian functional is also calculated. The gaussian

for *<sup>ξ</sup>* <sup>∈</sup> **<sup>C</sup>** with Re(*ξ*) <sup>&</sup>gt; 0. The double infinitesimal Fourier transform is calculated as

Finally a sort of functional is constructed in the theory that associates to Riemann's zeta function. The path integral is defined for the application in the theory, and it is shown that the path integral of the functional *Zs* corresponds to Riemann's zeta function in the case that Re(*s*) > 1. By using the Poisson summation formula for the functional, a relationship

The usual universe is extended, in order to treat many stages of delta functions and functions on the space of functions. For the extension, there exists two methods, one is the second extension of the universe in the nonstandard analysis ([8],[9]) and the other is the Relative set theory in the axiomatic set theory ( [13]). The first one ([8],[9]) is explained here, by using

Let <sup>Λ</sup> be an infinite set. Let *<sup>F</sup>* be a nonprincipal ultrafilter on <sup>Λ</sup>. For each *<sup>λ</sup>* <sup>∈</sup> <sup>Λ</sup>, let *<sup>S</sup><sup>λ</sup>* be a set. An equivalence relation <sup>∼</sup> is induced from *<sup>F</sup>* on <sup>∏</sup>*λ*∈<sup>Λ</sup> *<sup>S</sup>λ*. For *<sup>α</sup>* = (*αλ*), *<sup>β</sup>* = (*βλ*) (*<sup>λ</sup>* <sup>∈</sup>

The set of equivalence classes is called *ultraproduct* of *<sup>S</sup><sup>λ</sup>* for *<sup>F</sup>* with respect to <sup>∼</sup>. If *<sup>S</sup><sup>λ</sup>* <sup>=</sup> *<sup>S</sup>* for *<sup>λ</sup>* <sup>∈</sup> <sup>Λ</sup>, then it is called *ultraproduct* of *<sup>S</sup>* for *<sup>F</sup>* and it is written as <sup>∗</sup>*S*. The set *<sup>S</sup>* is naturally

where [ ] denotes the equivalence class with respect to the ultrafilter *F*. The mapping is written as <sup>∗</sup>, and call it naturally elementary embedding. From now on, we identify the

Let *<sup>H</sup>* (<sup>∈</sup> <sup>∗</sup>**Z**) be an infinite even number. The infinite number *<sup>H</sup>* is even, when for *<sup>H</sup>* <sup>=</sup>

infinitesimal lattice space **L**, an infinitesimal lattice subspace *L* and a space of functions *R*(*L*)

for *<sup>α</sup>* <sup>∈</sup> *<sup>L</sup>*2(**R**), in which *<sup>C</sup><sup>ξ</sup>* is a constant independent of *<sup>α</sup>* .

*<sup>α</sup>* <sup>∼</sup> *<sup>β</sup>* ⇐⇒ {*<sup>λ</sup>* <sup>∈</sup> <sup>Λ</sup> <sup>|</sup> *αλ* <sup>=</sup> *βλ*} ∈ *<sup>F</sup>*. (1)

*<sup>s</sup>* (<sup>∈</sup> *<sup>S</sup>*) �→ [(*s<sup>λ</sup>* <sup>=</sup> *<sup>s</sup>*), *<sup>λ</sup>* <sup>∈</sup> <sup>Λ</sup>] (<sup>∈</sup> <sup>∗</sup>*S*) (2)

*<sup>H</sup>* is written as *<sup>ε</sup>*. We define an

<sup>−</sup>*πξ* <sup>∞</sup>

<sup>−</sup><sup>∞</sup> *<sup>α</sup>*2(*t*)*dt*

 ,

functional means that the standard part of the image for *<sup>α</sup>* <sup>∈</sup> *<sup>L</sup>*<sup>2</sup> is exp

For *<sup>ϕ</sup>*, *<sup>ψ</sup>* <sup>∈</sup> *<sup>R</sup>*(*L*), the infinitesimal *<sup>δ</sup>* function, the infinitesimal Fourier transformation *<sup>F</sup><sup>ϕ</sup>* (<sup>∈</sup> *<sup>R</sup>*(*L*)), the inverse infinitesimal Fourier transformation *<sup>F</sup><sup>ϕ</sup>* (<sup>∈</sup> *<sup>R</sup>*(*L*)) and the convolution *<sup>ϕ</sup>* <sup>∗</sup> *<sup>ψ</sup>* (<sup>∈</sup> *<sup>R</sup>*(*L*)) are defined as follows :

$$\delta \in \mathcal{R}(L), \quad \delta(\mathbf{x}) := \begin{cases} H & (\mathbf{x} = \mathbf{0}) \\ \mathbf{0} & (\mathbf{x} \neq \mathbf{0}) \end{cases} \tag{3}$$

$$(F\wp)(p) := \sum\_{\mathbf{x} \in L} \varepsilon \exp\left(-2\pi i p \mathbf{x}\right) \wp(\mathbf{x})\tag{4}$$

$$(\overline{F}\varphi)(p) := \sum\_{\mathbf{x} \in L} \varepsilon \exp\left(2\pi i p \mathbf{x}\right) \varphi(\mathbf{x})\tag{5}$$

$$(\mathfrak{q}\*\psi)(\mathfrak{x}) := \sum\_{\mathfrak{y}\in L} \varepsilon \mathfrak{q}(\mathfrak{x}-\mathfrak{y})\psi(\mathfrak{y}).\tag{6}$$

The definition implies the following theorem as same as the Fourier transform for the finite discrete abelian group.

#### **Theorem 2.1.3.**


The definition implies the following proposition by the simple calculation:

#### **Proposition 2.1.4.**

If *<sup>l</sup>* <sup>∈</sup> **<sup>R</sup>**, then

$$F\delta^l = (H)^{(l-1)}.\tag{7}$$

**Examples of the infinitesimal Fourier transform for functions**

The infinitesimal Fourier transforms of the gaussian function *ϕξ* , *<sup>ϕ</sup>im* <sup>∈</sup> *<sup>R</sup>*(*L*) are calculated as follows: *ϕξ* (*x*) = exp(−*ξπx*2), where *<sup>ξ</sup>* <sup>∈</sup> **<sup>C</sup>**, Re(*ξ*) <sup>&</sup>gt; 0,

$$\varphi\_{im}(\mathbf{x}) = \exp(-im\pi\mathbf{x}^2)\_\prime \text{ where } m \in \mathbf{Z}.$$

For *ϕξ* , we obtain :

#### **Proposition 2.1.5.**

(*Fϕξ* )(*p*) = *c<sup>ξ</sup>* (*p*)*ϕξ* ( *<sup>p</sup> <sup>ξ</sup>* ), where *<sup>c</sup><sup>ξ</sup>* (*p*) = <sup>∑</sup>*x*∈*<sup>L</sup> <sup>ε</sup>* exp(−*ξπ*(*<sup>x</sup>* <sup>+</sup> *<sup>i</sup> <sup>ξ</sup> p*)2) and *p* is an element of the lattice *L*.

If *<sup>p</sup>* is finite, then st(*c<sup>ξ</sup>* (*p*)) = <sup>√</sup> 1 *ξ* .

**Proof.** The infinitesimal Fourier transforms of *ϕξ* is :

$$\begin{split} (F\rho\_{\xi})(p) &= \sum\_{\mathbf{x}\in L} \varepsilon \exp(-2\pi i p \mathbf{x}) \exp(-\xi \pi \mathbf{x}^{2}) \\ &= (\sum\_{\mathbf{x}\in L} \varepsilon \exp(-\xi \pi (\mathbf{x} + \frac{i}{\xi}p)^{2})) \exp(-\pi \frac{1}{\xi} p^{2}) = c\_{\xi}(p) \varphi\_{\xi}(\frac{p}{\xi}) \end{split} \tag{8}$$

where *<sup>c</sup><sup>ξ</sup>* (*p*) = <sup>∑</sup>*x*∈*<sup>L</sup> <sup>ε</sup>* exp(−*ξπ*(*<sup>x</sup>* <sup>+</sup> *<sup>i</sup> <sup>ξ</sup> p*)2). If *p* is finite, then st(*c<sup>ξ</sup>* (*p*)) = <sup>∞</sup> <sup>−</sup><sup>∞</sup> exp <sup>−</sup>*ξπ t* + *<sup>i</sup> <sup>ξ</sup>* st(*p*) 2 *dt* = <sup>√</sup> 1 *ξ* .

Theorem 2.1.3 (8) implies the following for *cξ* :

**Proposition 2.1.6.**

$$
\varphi\_{\xi}(\mathbf{x}') = \left(\overline{F}c\_{\xi}(p) \* \left(c\_{\frac{1}{\xi}}(-\mathbf{x})\varphi\_{\xi}(\mathbf{x})\right)\right)(\mathbf{x}').\tag{9}
$$

**Proof.** It is obtained : (*Fϕξ* )(*p*) = *c<sup>ξ</sup>* (*p*)*ϕξ* ( *<sup>p</sup> <sup>ξ</sup>* ), and put *F* to the above : (*F*(*Fϕξ* ))(*x*)=(*F*(*c<sup>ξ</sup>* (*p*)*ϕξ* ( *<sup>p</sup> <sup>ξ</sup>* )))(*x*) = (*Fc<sup>ξ</sup>* (*p*) <sup>∗</sup> *<sup>F</sup>ϕξ* ( *<sup>p</sup> <sup>ξ</sup>* ))(*x*), that is, *ϕξ* (*x*)=(*Fc<sup>ξ</sup>* (*p*) <sup>∗</sup> *<sup>F</sup>ϕξ* ( *<sup>p</sup> <sup>ξ</sup>* ))(*x*). Now (*Fϕξ* ( *<sup>p</sup> <sup>ξ</sup>* ))(*x*) = <sup>∑</sup>*p*∈*<sup>L</sup> <sup>ε</sup>* exp(−2*πipx*) exp(−*ξ*( *<sup>p</sup> <sup>ξ</sup>* )2*π*) <sup>=</sup> <sup>∑</sup>*p*∈*<sup>L</sup> <sup>ε</sup>* exp(−*<sup>π</sup>* <sup>1</sup> *<sup>ξ</sup>* (*p*<sup>2</sup> <sup>−</sup> <sup>2</sup>*πi<sup>ξ</sup> px*)) = <sup>∑</sup>*p*∈*<sup>L</sup> <sup>ε</sup>* exp(−*<sup>π</sup> <sup>ξ</sup>* (*<sup>p</sup>* <sup>−</sup> *<sup>i</sup>ξx*)2) *ϕξ* (*x*). By the definition : *<sup>c</sup><sup>ξ</sup>* (*p*) = <sup>∑</sup>*x*∈*<sup>L</sup> <sup>ε</sup>* exp(−*πξ*(*<sup>x</sup>* <sup>+</sup> *<sup>i</sup>* <sup>1</sup> *<sup>ξ</sup> p*)2), the sum <sup>∑</sup>*p*∈*<sup>L</sup> <sup>ε</sup>* exp(−*<sup>π</sup> <sup>ξ</sup>* (*<sup>p</sup>* <sup>−</sup> *<sup>i</sup>ξx*)2) is *<sup>c</sup>* <sup>1</sup> *ξ* (−*x*). Hence

$$\varphi\_{\vec{\xi}}(\mathbf{x}') = \left(\overline{\mathcal{F}}c\_{\vec{\xi}}(p) \* \left(c\_{\frac{1}{\xi}}(-\mathbf{x})\varphi\_{\vec{\xi}}(\mathbf{x})\right)\right)(\mathbf{x}').\tag{10}$$

For the following proposition 2.1.7, the Gauss sum is recalled (cf.[15]) : For *<sup>z</sup>* <sup>∈</sup> **<sup>N</sup>**, the Gauss sum ∑*z*−<sup>1</sup> *<sup>l</sup>*=<sup>0</sup> exp(−*<sup>i</sup>* <sup>2</sup>*<sup>π</sup> z l* <sup>2</sup>) is equal to √*z* <sup>1</sup>+(−*i*)*<sup>z</sup>* <sup>1</sup>−*<sup>i</sup>* .

**Proposition 2.1.7.** If *<sup>m</sup>*|2*H*<sup>2</sup> and *<sup>m</sup>*<sup>|</sup> *p ε* , then

4 ime knjige

The infinitesimal Fourier transforms of the gaussian function *ϕξ* , *<sup>ϕ</sup>im* <sup>∈</sup> *<sup>R</sup>*(*L*) are calculated

*<sup>ξ</sup>* ), where *<sup>c</sup><sup>ξ</sup>* (*p*) = <sup>∑</sup>*x*∈*<sup>L</sup> <sup>ε</sup>* exp(−*ξπ*(*<sup>x</sup>* <sup>+</sup> *<sup>i</sup>*

*<sup>ε</sup>* exp(−2*πipx*) exp(−*ξπx*2)

*i*

*<sup>ξ</sup> <sup>p</sup>*)2)) exp(−*<sup>π</sup>*

*<sup>ξ</sup> p*)2). If *p* is finite, then st(*c<sup>ξ</sup>* (*p*))

(−*x*)*ϕξ* (*x*)

*<sup>ξ</sup>* )2*π*)

1

 (*x*′

*<sup>ξ</sup>* ), and put *F* to the above :

*<sup>ξ</sup>* ))(*x*).

*<sup>ξ</sup>* (*<sup>p</sup>* <sup>−</sup> *<sup>i</sup>ξx*)2)

 (*x*′

*<sup>ξ</sup> p*)2), the sum

(−*x*)*ϕξ* (*x*)

 *ϕξ* (*x*).

*<sup>ξ</sup> <sup>p</sup>*2) = *<sup>c</sup><sup>ξ</sup>* (*p*)*ϕξ* (

*<sup>ε</sup>* exp(−*ξπ*(*<sup>x</sup>* <sup>+</sup>

*dt* = <sup>√</sup> 1 *ξ* .

*Fc<sup>ξ</sup>* (*p*) <sup>∗</sup>

 *c* 1 *ξ*

<sup>∑</sup>*p*∈*<sup>L</sup> <sup>ε</sup>* exp(−*<sup>π</sup>*

 *c* 1 *ξ* *<sup>ξ</sup> p*)2) and *p* is an element of

*p ξ*

). (9)

). (10)

) (8)

as follows: *ϕξ* (*x*) = exp(−*ξπx*2), where *<sup>ξ</sup>* <sup>∈</sup> **<sup>C</sup>**, Re(*ξ*) <sup>&</sup>gt; 0,

1 *ξ* .

**Proof.** The infinitesimal Fourier transforms of *ϕξ* is :

*<sup>x</sup>*∈*<sup>L</sup>*

= (∑ *<sup>x</sup>*∈*<sup>L</sup>*

Theorem 2.1.3 (8) implies the following for *cξ* :

**Proof.** It is obtained : (*Fϕξ* )(*p*) = *c<sup>ξ</sup>* (*p*)*ϕξ* ( *<sup>p</sup>*

*ϕξ* (*x*′ ) = 

*<sup>ξ</sup>* )))(*x*)

*<sup>ξ</sup>* ))(*x*) = <sup>∑</sup>*p*∈*<sup>L</sup> <sup>ε</sup>* exp(−2*πipx*) exp(−*ξ*( *<sup>p</sup>*

*ξ*

*ϕξ* (*x*′ ) = 

*<sup>ξ</sup>* (*p*<sup>2</sup> <sup>−</sup> <sup>2</sup>*πi<sup>ξ</sup> px*)) =

By the definition : *<sup>c</sup><sup>ξ</sup>* (*p*) = <sup>∑</sup>*x*∈*<sup>L</sup> <sup>ε</sup>* exp(−*πξ*(*<sup>x</sup>* <sup>+</sup> *<sup>i</sup>* <sup>1</sup>

*<sup>ξ</sup>* (*<sup>p</sup>* <sup>−</sup> *<sup>i</sup>ξx*)2) is *<sup>c</sup>* <sup>1</sup>

*<sup>ξ</sup>* ))(*x*), that is, *ϕξ* (*x*)=(*Fc<sup>ξ</sup>* (*p*) <sup>∗</sup> *<sup>F</sup>ϕξ* ( *<sup>p</sup>*

(−*x*). Hence

*Fc<sup>ξ</sup>* (*p*) <sup>∗</sup>

*<sup>ϕ</sup>im*(*x*) = exp(−*imπx*2), where *<sup>m</sup>* <sup>∈</sup> **<sup>Z</sup>**.

If *<sup>p</sup>* is finite, then st(*c<sup>ξ</sup>* (*p*)) = <sup>√</sup>

(*Fϕξ* )(*p*) = ∑

where *<sup>c</sup><sup>ξ</sup>* (*p*) = <sup>∑</sup>*x*∈*<sup>L</sup> <sup>ε</sup>* exp(−*ξπ*(*<sup>x</sup>* <sup>+</sup> *<sup>i</sup>*

 <sup>−</sup>*ξπ t* + *<sup>i</sup> <sup>ξ</sup>* st(*p*) 2 

(*F*(*Fϕξ* ))(*x*)=(*F*(*c<sup>ξ</sup>* (*p*)*ϕξ* ( *<sup>p</sup>*

**Proposition 2.1.6.**

= (*Fc<sup>ξ</sup>* (*p*) <sup>∗</sup> *<sup>F</sup>ϕξ* ( *<sup>p</sup>*

<sup>=</sup> <sup>∑</sup>*p*∈*<sup>L</sup> <sup>ε</sup>* exp(−*<sup>π</sup>* <sup>1</sup>

<sup>∑</sup>*p*∈*<sup>L</sup> <sup>ε</sup>* exp(−*<sup>π</sup>*

Now (*Fϕξ* ( *<sup>p</sup>*

For *ϕξ* , we obtain : **Proposition 2.1.5.** (*Fϕξ* )(*p*) = *c<sup>ξ</sup>* (*p*)*ϕξ* ( *<sup>p</sup>*

the lattice *L*.

= <sup>∞</sup> <sup>−</sup><sup>∞</sup> exp

$$(F\varphi\_{im})(p) = c\_{im}(p)\exp(i\pi\frac{1}{m}p^2) \tag{11}$$

where *cim*(*p*) = *<sup>m</sup>* 2 1+*i* 2*H*2 *m* <sup>1</sup>+*<sup>i</sup>* for positive *<sup>m</sup>* and *cim*(*p*) = <sup>−</sup>*<sup>m</sup>* 2 <sup>1</sup>+(−*i*) 2*H*2 −*m* <sup>1</sup>−*<sup>i</sup>* for negative *<sup>m</sup>*. **Proof.** (*Fϕim*)(*p*) = <sup>∑</sup>*x*∈*<sup>L</sup> <sup>ε</sup>* exp(−*imπx*2) exp(−2*πixp*) = *cim*(*p*) exp(*iπ* <sup>1</sup> *<sup>m</sup> <sup>p</sup>*2), where *cim*(*p*) = <sup>∑</sup>*x*∈*<sup>L</sup> <sup>ε</sup>* exp(−*imπ*(*<sup>x</sup>* <sup>+</sup> *<sup>p</sup> <sup>m</sup>* )2).

Since *<sup>m</sup>*<sup>|</sup> *p <sup>ε</sup>* , the element *<sup>p</sup> <sup>m</sup>* is in *<sup>L</sup>*. It is remarked that exp(−*iπmx*2) = exp(−*iπm*(*<sup>x</sup>* <sup>+</sup> *<sup>H</sup>*)2). For positive *m*,

$$c\_{im}(p) = \sum\_{\mathbf{x} \in L} \varepsilon \exp(-im\pi\mathbf{x}^2) = \frac{m}{2} \overline{\left(\varepsilon \sqrt{\frac{2H^2}{m}} \frac{1 + (-i)^{\frac{2H^2}{m}}}{1 - i}\right)}\tag{12}$$

by the above Gauss sum. Hence *cim*(*p*) = *<sup>m</sup>* 2 1+*i* 2*H*2 *m* <sup>1</sup>+*<sup>i</sup>* . For negative *m*, the proof is as same as the above.

#### **2.2. Second extension of the universe**

To treat a ∗-unbounded functional *f* in the nonstandard analysis, we need a second nonstandardization. Let *F*<sup>2</sup> := *F* be a nonprincipal ultrafilter on an infinite set Λ<sup>2</sup> := Λ as above. Denote the ultraproduct of a set *S* with respect to *F*<sup>2</sup> by ∗*S* as above. Let *F*<sup>1</sup> be another nonprincipal ultrafilter on an infinite set Λ1. Take the <sup>∗</sup>-ultrafilter <sup>∗</sup>*F*<sup>1</sup> on <sup>∗</sup>Λ1. For an internal set *<sup>S</sup>* in the sense of <sup>∗</sup>-nonstandardization, let <sup>⋆</sup>*<sup>S</sup>* be the <sup>∗</sup>-ultraproduct of *<sup>S</sup>* with respect to <sup>∗</sup>*F*1. Thus, a double ultraproduct <sup>⋆</sup>(∗**R**), <sup>⋆</sup>(∗**Z**), etc are defined for the set **<sup>R</sup>**, **<sup>Z</sup>**, etc. It is shown easily that

$$\mathbf{S^{\star}}(^{\ast}\mathbf{S}) = \mathbf{S^{\Lambda\_1 \times \Lambda\_2}} / F\_1^{F\_2} \, \, \, \, \tag{13}$$

where *FF*<sup>2</sup> <sup>1</sup> denotes the ultrafilter on <sup>Λ</sup><sup>1</sup> <sup>×</sup> <sup>Λ</sup><sup>2</sup> such that for any *<sup>A</sup>* <sup>⊂</sup> <sup>Λ</sup><sup>1</sup> <sup>×</sup> <sup>Λ</sup>2, *<sup>A</sup>* <sup>∈</sup> *<sup>F</sup>F*<sup>2</sup> <sup>1</sup> if and only if

$$\{\lambda \in \Lambda\_1 \mid \{\mu \in \Lambda\_2 \mid (\lambda, \mu) \in A\} \in F\_2\} \in F\_1. \tag{14}$$

The work is done with this double nonstandardization. The natural imbedding <sup>⋆</sup>*<sup>S</sup>* of an internal element *S* which is not considered as a set in ∗-nonstandardization is often denoted simply by *S*.

An infinite number in <sup>⋆</sup>(∗**R**) is defined to be greater than any element in <sup>∗</sup>**R**. We remark that an infinite number in <sup>∗</sup>**<sup>R</sup>** is not infinite in <sup>⋆</sup>( <sup>∗</sup>**R**), that is, the word ′′an infinite number in <sup>⋆</sup>(∗**R**)′′ has a double meaning. An infinitesimal number in <sup>⋆</sup>(∗**R**) is also defined to be nonzero and whose absolute value is less than each positive number in <sup>∗</sup>**R**.

#### **Definition 2.2.1.**

Let *<sup>H</sup>*(<sup>∈</sup> <sup>∗</sup>**Z**), *<sup>H</sup>*′ (<sup>∈</sup> <sup>⋆</sup>(∗**Z**)) be even positive numbers such that *<sup>H</sup>*′ is larger than any element in <sup>∗</sup>**Z**, and let *<sup>ε</sup>*(<sup>∈</sup> <sup>∗</sup>**R**), *<sup>ε</sup>*′ (<sup>∈</sup> <sup>⋆</sup>(∗**R**)) be infinitesimals satifying *<sup>ε</sup><sup>H</sup>* <sup>=</sup> 1, *<sup>ε</sup>*′ *H*′ = 1. We define as follows :

$$\begin{split} \mathbf{L} & \coloneqq \varepsilon^\* \mathbf{Z} = \{ \varepsilon z \, \big|\, z \in \, ^\* \mathbf{Z} \}, \; \mathbf{L}' := \varepsilon^{\prime \*} (\, ^\* \mathbf{Z}) = \{ \varepsilon^{\prime} z^{\prime} \, \big|\, z' \in \, ^\* (\, ^\* \mathbf{Z}) \}, \\ \mathcal{L} & \coloneqq \left\{ \varepsilon z \, \bigg|\, z \in \, ^\* \mathbf{Z} \, -\frac{H}{2} \le \varepsilon z < \frac{H}{2} \right\} (\subset \mathbf{L}), \\ \mathcal{L}' & \coloneqq \left\{ \varepsilon^{\prime} z^{\prime} \, \bigg|\, z^{\prime} \in \, ^\* (\, ^\* \mathbf{Z}) \, -\frac{H^{\prime}}{2} \le \varepsilon^{\prime} z^{\prime} < \frac{H^{\prime}}{2} \right\} (\subset \mathbf{L}'). \end{split}$$

Here *L* is an ultraproduct of lattices

 $L\_{\mu} := \left\{ \varepsilon\_{\mu} z\_{\mu} \, \middle| \, z\_{\mu} \in \mathbf{Z}, -\frac{H\_{\mu}}{2} \le \varepsilon\_{\mu} z\_{\mu} < \frac{H\_{\mu}}{2} \right\} \text{ ( $\mu \in \Lambda\_{2}$ )}$ 

in **R**, and *L*′ is also an ultraproduct of lattices

$$L'\_{\lambda} := \left\{ \varepsilon'\_{\lambda} z'\_{\lambda} \, \middle| \, z'\_{\lambda} \in \, ^\*\mathbf{Z}\_{\prime} - \frac{H'\_{\lambda}}{2} \le \varepsilon'\_{\lambda} z'\_{\lambda} < \frac{H'\_{\lambda}}{2} \right\} \, (\lambda \in \Lambda\_1),$$

in <sup>∗</sup>**R** that is an ultraproduct of

$$L'\_{\lambda\mu} := \left\{ \varepsilon'\_{\lambda\mu} z'\_{\lambda\mu} \, \middle| \, z'\_{\lambda\mu} \in \mathbf{Z}\_\prime - \frac{H'\_{\lambda\mu}}{2} \le \varepsilon'\_{\lambda\mu} z'\_{\lambda\mu} < \frac{H'\_{\lambda\mu}}{2} \right\} \ (\mu \in \Lambda\_2).$$

A latticed space of functions *X* is defined as follows,

$$\begin{aligned} X &:= \{ a \mid a \text{ is an internal function with double meanings, from } ^\star L \text{ to } L' \} \\ &= \{ [(a\_\lambda), \lambda \in \Lambda\_1] \mid a\_\lambda \text{ is an internal function from } L \text{ to } L'\_\lambda \} \end{aligned} \tag{15}$$

where *<sup>a</sup><sup>λ</sup>* : *<sup>L</sup>* <sup>→</sup> *<sup>L</sup>*′ *<sup>λ</sup>* is *<sup>a</sup><sup>λ</sup>* = [(*aλµ*), *<sup>µ</sup>* <sup>∈</sup> <sup>Λ</sup>2], *<sup>a</sup>λµ* : *<sup>L</sup><sup>µ</sup>* <sup>→</sup> *<sup>L</sup>*′ *λµ*.

represented as the following internal set :

Three equivalence relations <sup>∼</sup>*H*, <sup>∼</sup>⋆(*H*) and <sup>∼</sup>*H*′ are defined on **<sup>L</sup>**, <sup>⋆</sup>(**L**) and **<sup>L</sup>**′ :

$$\mathbf{x} \sim\_H \mathbf{y} \Longleftrightarrow \mathbf{x} - y \in H^\* \mathbf{Z}, \ x \sim\_{\star(H)} y \Longleftrightarrow \mathbf{x} - y \in \star(H)^\*(^\*\mathbf{Z}),$$

*<sup>x</sup>* <sup>∼</sup>*H*′ *<sup>y</sup>* ⇐⇒ *<sup>x</sup>* <sup>−</sup> *<sup>y</sup>* <sup>∈</sup> *<sup>H</sup>*′ <sup>⋆</sup>( <sup>∗</sup>**Z**). Then **<sup>L</sup>**/ <sup>∼</sup>*H*, <sup>⋆</sup>(**L**)/ <sup>∼</sup>⋆(*H*) and **<sup>L</sup>**′ / <sup>∼</sup>*H*′ are identified as *<sup>L</sup>*, <sup>⋆</sup>(*L*) and *<sup>L</sup>*′ is identified with *<sup>L</sup>*, the set <sup>⋆</sup>(**L**)/ <sup>∼</sup>⋆(*H*) is identified with **<sup>L</sup>**/ <sup>∼</sup>*H*. Furthermore *<sup>X</sup>* is

$$\{a \mid a \text{ is an internal function from } \star(\mathbf{L})/\sim\_{\star(H)} \text{ to } \mathbf{L}'/\sim\_{H'}\}.\tag{16}$$

. Since <sup>⋆</sup>(*L*)

The same notation is used as a function from <sup>⋆</sup>(*L*) to *<sup>L</sup>*′ to represent a function in the above internal set. The space *A* of functionals is defined as follows:

*<sup>A</sup>* :<sup>=</sup> { *<sup>f</sup>* <sup>|</sup> *<sup>f</sup>* is an internal function with a double meaning from *<sup>X</sup>* to <sup>⋆</sup>( <sup>∗</sup>**C**)}. (17)

An infinitesimal delta function *<sup>δ</sup>*(*a*)(<sup>∈</sup> *<sup>A</sup>*), an infinitesimal Fourier transform of *<sup>f</sup>*(<sup>∈</sup> *<sup>A</sup>*), an inverse infinitesimal Fourier transform of *<sup>f</sup>* and a convolution of *<sup>f</sup>* , *<sup>g</sup>*(<sup>∈</sup> *<sup>A</sup>*), are defined by the following :

**Definition 2.2.2.** The delta function

6 ime knjige

**Definition 2.2.1.**

Let *<sup>H</sup>*(<sup>∈</sup> <sup>∗</sup>**Z**), *<sup>H</sup>*′

as follows :

*L* := *εz* 

*L*′ := *ε*′ *z*′ 

*Lµ* := *εµz<sup>µ</sup>* 

*L*′ *<sup>λ</sup>* := *ε*′ *<sup>λ</sup>z*′ *λ z*′

*L*′ *λµ* := *ε*′ *λµz*′ *λµ z*′

where *<sup>a</sup><sup>λ</sup>* : *<sup>L</sup>* <sup>→</sup> *<sup>L</sup>*′

in <sup>∗</sup>**Z**, and let *<sup>ε</sup>*(<sup>∈</sup> <sup>∗</sup>**R**), *<sup>ε</sup>*′

*<sup>z</sup>* <sup>∈</sup> <sup>∗</sup>**Z**, <sup>−</sup> *<sup>H</sup>*

Here *L* is an ultraproduct of lattices

in <sup>∗</sup>**R** that is an ultraproduct of

*<sup>x</sup>* <sup>∼</sup>*H*′ *<sup>y</sup>* ⇐⇒ *<sup>x</sup>* <sup>−</sup> *<sup>y</sup>* <sup>∈</sup> *<sup>H</sup>*′ <sup>⋆</sup>( <sup>∗</sup>**Z**).

Then **<sup>L</sup>**/ <sup>∼</sup>*H*, <sup>⋆</sup>(**L**)/ <sup>∼</sup>⋆(*H*) and **<sup>L</sup>**′

represented as the following internal set :

**<sup>L</sup>** :<sup>=</sup> *<sup>ε</sup>* <sup>∗</sup>**<sup>Z</sup>** <sup>=</sup> {*ε<sup>z</sup>* <sup>|</sup> *<sup>z</sup>* <sup>∈</sup> <sup>∗</sup>**Z**}, **<sup>L</sup>**′ :<sup>=</sup> *<sup>ε</sup>*′ <sup>⋆</sup>( <sup>∗</sup>**Z**) = {*ε*′

*<sup>z</sup>*′ <sup>∈</sup> <sup>⋆</sup>( <sup>∗</sup>**Z**), <sup>−</sup> *<sup>H</sup>*′

*<sup>z</sup><sup>µ</sup>* <sup>∈</sup> **<sup>Z</sup>**, <sup>−</sup> *<sup>H</sup><sup>µ</sup>*

in **R**, and *L*′ is also an ultraproduct of lattices

*<sup>λ</sup>* <sup>∈</sup> <sup>∗</sup>**Z**, <sup>−</sup> *<sup>H</sup>*′

*λµ* <sup>∈</sup> **<sup>Z</sup>**, <sup>−</sup> *<sup>H</sup>*′

A latticed space of functions *X* is defined as follows,

<sup>2</sup> <sup>≤</sup> *<sup>ε</sup><sup>z</sup>* <sup>&</sup>lt; *<sup>H</sup>*

2 

<sup>2</sup> <sup>≤</sup> *<sup>ε</sup>*′

<sup>2</sup> <sup>≤</sup> *εµz<sup>µ</sup>* <sup>&</sup>lt;

*λµ* <sup>2</sup> <sup>≤</sup> *<sup>ε</sup>*′

*λ* <sup>2</sup> <sup>≤</sup> *<sup>ε</sup>*′ *<sup>λ</sup>z*′ *<sup>λ</sup>* <sup>&</sup>lt; *<sup>H</sup>*′ *λ* 2 

An infinite number in <sup>⋆</sup>(∗**R**) is defined to be greater than any element in <sup>∗</sup>**R**. We remark that an infinite number in <sup>∗</sup>**<sup>R</sup>** is not infinite in <sup>⋆</sup>( <sup>∗</sup>**R**), that is, the word ′′an infinite number in <sup>⋆</sup>(∗**R**)′′ has a double meaning. An infinitesimal number in <sup>⋆</sup>(∗**R**) is also defined to be

(<sup>∈</sup> <sup>⋆</sup>(∗**Z**)) be even positive numbers such that *<sup>H</sup>*′ is larger than any element

*<sup>z</sup>*′ <sup>|</sup> *<sup>z</sup>*′ <sup>∈</sup> <sup>⋆</sup>( <sup>∗</sup>**Z**)},

*H*′ = 1. We define

}

*<sup>λ</sup>*} (15)

. Since <sup>⋆</sup>(*L*)

/ <sup>∼</sup>*H*′ }. (16)

(<sup>∈</sup> <sup>⋆</sup>(∗**R**)) be infinitesimals satifying *<sup>ε</sup><sup>H</sup>* <sup>=</sup> 1, *<sup>ε</sup>*′

(*<sup>µ</sup>* <sup>∈</sup> <sup>Λ</sup>2)

(*<sup>λ</sup>* <sup>∈</sup> <sup>Λ</sup>1)

*<sup>X</sup>* :<sup>=</sup> {*<sup>a</sup>* <sup>|</sup> *<sup>a</sup>* is an internal function with double meanings, from <sup>⋆</sup>*<sup>L</sup>* to *<sup>L</sup>*′

is identified with *<sup>L</sup>*, the set <sup>⋆</sup>(**L**)/ <sup>∼</sup>⋆(*H*) is identified with **<sup>L</sup>**/ <sup>∼</sup>*H*. Furthermore *<sup>X</sup>* is

{*<sup>a</sup>* <sup>|</sup> *<sup>a</sup>* is an internal function from <sup>⋆</sup>(**L**)/ <sup>∼</sup>⋆(*H*) to **<sup>L</sup>**′

<sup>=</sup> {[(*aλ*), *<sup>λ</sup>* <sup>∈</sup> <sup>Λ</sup>1] <sup>|</sup> *<sup>a</sup><sup>λ</sup>* is an internal function from *<sup>L</sup>* to *<sup>L</sup>*′

(*<sup>µ</sup>* <sup>∈</sup> <sup>Λ</sup>2).

*λµ*.

/ <sup>∼</sup>*H*′ are identified as *<sup>L</sup>*, <sup>⋆</sup>(*L*) and *<sup>L</sup>*′

nonzero and whose absolute value is less than each positive number in <sup>∗</sup>**R**.

(<sup>⊂</sup> **<sup>L</sup>**),

*<sup>z</sup>*′ < *<sup>H</sup>*′ 2 (<sup>⊂</sup> **<sup>L</sup>**′ ).

> *Hµ* 2

*λµz*′ *λµ* < *H*′ *λµ* 2 

*<sup>λ</sup>* is *<sup>a</sup><sup>λ</sup>* = [(*aλµ*), *<sup>µ</sup>* <sup>∈</sup> <sup>Λ</sup>2], *<sup>a</sup>λµ* : *<sup>L</sup><sup>µ</sup>* <sup>→</sup> *<sup>L</sup>*′

*<sup>x</sup>* <sup>∼</sup>*<sup>H</sup> <sup>y</sup>* ⇐⇒ *<sup>x</sup>* <sup>−</sup> *<sup>y</sup>* <sup>∈</sup> *<sup>H</sup>* <sup>∗</sup>**Z**, *<sup>x</sup>* <sup>∼</sup>⋆(*H*) *<sup>y</sup>* ⇐⇒ *<sup>x</sup>* <sup>−</sup> *<sup>y</sup>* <sup>∈</sup> <sup>⋆</sup>(*H*) <sup>⋆</sup>( <sup>∗</sup>**Z**),

Three equivalence relations <sup>∼</sup>*H*, <sup>∼</sup>⋆(*H*) and <sup>∼</sup>*H*′ are defined on **<sup>L</sup>**, <sup>⋆</sup>(**L**) and **<sup>L</sup>**′ :

$$\delta(a) := \begin{cases} (H')^{("H")^2} & (a=0) \\ 0 & (a \neq 0) \end{cases} \tag{18}$$

and, with *ε*<sup>0</sup> := (*H*′ )−( <sup>⋆</sup>*H*)<sup>2</sup> <sup>∈</sup> <sup>⋆</sup>(∗**R**),

$$(\!(Ff))(b) := \sum\_{a \in X} \varepsilon\_0 \exp\left(-2\pi i \sum\_{k \in L} a(k)b(k)\right) f(a) \tag{19}$$

$$(\overline{F}f)(b) := \sum\_{a \in X} \varepsilon\_0 \exp\left(2\pi i \sum\_{k \in L} a(k)b(k)\right) f(a) \tag{20}$$

$$(f\*g)(a) := \sum\_{a' \in X} \epsilon\_0 f(a - a')g(a'). \tag{21}$$

The inner product on *A* is defined as:

$$\mathfrak{g}(f,\mathfrak{g}) := \sum\_{b \in X} \varepsilon\_0 \overline{f(b)} \mathfrak{g}(b),\tag{22}$$

where *f*(*b*) is the complex conjugate of *f*(*b*). In the section 3, Riemann's zeta function is written down as a nonstandard functional in Definition 2.2.2. In general, <sup>∑</sup>*k*∈*<sup>L</sup> <sup>a</sup>*2(*k*) is infinite, and it is difficult to consider the meaining of *F*, *F* in Definition 2.2.2 as standard objects. They are defined only algebraically. In order to understand Definition 2.2.2 analytically for a standard one, we change the definition briefly, to Definition 2.2.3. By replacing the definitions of *L*′ , *δ*, *ε*0, *F*, *F* in Definition 2.2.2 as the following, another type of infinitesimal Fourier transformation is defined later. The different point is only the definition of an inner product of the space of functions *X*. In Definition 2.2.2 , the inner product of *<sup>a</sup>*, *<sup>b</sup>*(<sup>∈</sup> *<sup>X</sup>*) is <sup>∑</sup>*k*∈*<sup>L</sup> <sup>a</sup>*(*k*)*b*(*k*), and in the following definition, it is <sup>⋆</sup>*<sup>ε</sup>* <sup>∑</sup>*k*∈*<sup>L</sup> <sup>a</sup>*(*k*)*b*(*k*). **Definition 2.2.3.** *L*′ := *ε*′ *z*′ *<sup>z</sup>*′ <sup>∈</sup> <sup>⋆</sup>( <sup>∗</sup>**Z**), <sup>−</sup> <sup>⋆</sup>*<sup>H</sup> <sup>H</sup>*′ <sup>2</sup> <sup>≤</sup> *<sup>ε</sup>*′ *<sup>z</sup>*′ <sup>&</sup>lt; <sup>⋆</sup>*<sup>H</sup> <sup>H</sup>*′ 2 ,

$$\delta(a) := \begin{cases} (\,^\star H)^{\frac{(\,^\star H)^2}{2}} H^{\prime (\,^\star H)^2} & (a=0), \\ 0 & (a \neq 0) \end{cases} \tag{23}$$

and, with *<sup>ε</sup>*<sup>0</sup> := ( <sup>⋆</sup>*H*)<sup>−</sup> ( <sup>⋆</sup> *<sup>H</sup>*)<sup>2</sup> <sup>2</sup> *<sup>H</sup>*′−( <sup>⋆</sup>*H*)<sup>2</sup>

$$(\!(Ff))(b) := \sum\_{a \in X} \varepsilon\_0 \exp\left(-2\pi i \, ^\star \varepsilon \sum\_{k \in L} a(k)b(k)\right) f(a) \tag{24}$$

$$(\overline{F}f)(b) := \sum\_{a \in X} \varepsilon\_0 \exp\left(2\pi i \,^\star \varepsilon \sum\_{k \in L} a(k)b(k)\right) f(a). \tag{25}$$

Then the lattice *L*′ *λµ* is an abelian group for each *λµ*. The following theorem is obtained as same as the case of the discrete abelian group :

#### **Theorem 2.2.4.**

(1) *δ* = *F*1 = *F*1, (2) *F* is unitary, *F*<sup>4</sup> = 1, *FF* = *FF* = 1, (3) *<sup>f</sup>* <sup>∗</sup> *<sup>δ</sup>* <sup>=</sup> *<sup>δ</sup>* <sup>∗</sup> *<sup>f</sup>* <sup>=</sup> *<sup>f</sup>* , (4) *<sup>f</sup>* <sup>∗</sup> *<sup>g</sup>* <sup>=</sup> *<sup>g</sup>* <sup>∗</sup> *<sup>f</sup>* , (5) *<sup>F</sup>*(*<sup>f</sup>* <sup>∗</sup> *<sup>g</sup>*)=(*F f*)(*Fg*), (6) *<sup>F</sup>*(*<sup>f</sup>* <sup>∗</sup> *<sup>g</sup>*)=(*F f*)(*Fg*), (7) *<sup>F</sup>*(*f g*)=(*F f*) <sup>∗</sup> (*Fg*), (8) *<sup>F</sup>*(*f g*)=(*F f*) <sup>∗</sup> (*Fg*).

The definition directly implies the following proposition :

**Proposition 2.2.5.** If *<sup>l</sup>* <sup>∈</sup> **<sup>R</sup>**+, then

$$F\delta^l = (H')^{(l-1)("\,^\*H)^2}.\tag{26}$$

If there exists *<sup>α</sup>*, *<sup>β</sup>* <sup>∈</sup> *<sup>L</sup>*2(**R**) so that *<sup>a</sup>* <sup>=</sup> <sup>∗</sup>*α*|*L*, *<sup>b</sup>* <sup>=</sup> <sup>∗</sup>*β*|*L*, that is, *<sup>a</sup>*(*k*) = <sup>⋆</sup>( <sup>∗</sup>*α*(*k*)), *<sup>b</sup>*(*k*) = <sup>⋆</sup>( <sup>∗</sup>*β*(*k*)), then st(st(⋆*<sup>ε</sup>* <sup>∑</sup>*k*∈*<sup>L</sup> <sup>a</sup>*(*k*)*b*(*k*))) = <sup>∞</sup> <sup>−</sup><sup>∞</sup> *<sup>α</sup>*(*x*)*b*(*x*)*dx*. Definition 2.2.3 is easier understanding than Definition 2.2.2 for a standard meaning in analysis. For the reason, we consider mainly Definition 2.2.3 about several examples. However Definition 2.2.2 is also treated algebraically, as algebraically defined functions are not always *L*2-functions on **R**. The two types of Fourier transforms are different in a standard meaning.

#### **Examples of the double infinitesimal Fourier transform**

It is defined: an equivalence relation <sup>∼</sup><sup>⋆</sup>*HH*′ in **<sup>L</sup>**′ by *<sup>x</sup>* <sup>∼</sup><sup>⋆</sup>*HH*′ *<sup>y</sup>* <sup>⇔</sup> *<sup>x</sup>* <sup>−</sup> *<sup>y</sup>* <sup>∈</sup> <sup>⋆</sup>*HH*′ <sup>⋆</sup>( <sup>∗</sup>**Z**). The quotient space **L**′ / <sup>∼</sup><sup>⋆</sup>*HH*′ is defined with *<sup>L</sup>*′ . Let

*XH*, <sup>⋆</sup>*HH*′ :<sup>=</sup> {*a*′ <sup>|</sup> *<sup>a</sup>*′ is an internal function with a double meaning, from <sup>⋆</sup>**L**/ <sup>∼</sup>⋆(*H*) to **L**′ / <sup>∼</sup><sup>⋆</sup>*HH*′ }

and let **<sup>e</sup>** be a mapping from *<sup>X</sup>* to *XH*, <sup>⋆</sup>*HH*′ , defined by (**e**(*a*))([*k*]) = [*a*(<sup>ˆ</sup> *k*)], where [ ] on the left-hand side represents the equivalence class for the equivalence relation <sup>∼</sup>⋆(*H*) in <sup>⋆</sup>**L**, <sup>ˆ</sup> *k* is a representative in <sup>⋆</sup>(*L*) satisfying *<sup>k</sup>* <sup>∼</sup>⋆(*H*) <sup>ˆ</sup> *k*, and [ ] on the right-hand side represents the equivalence class for the equivalence relation <sup>∼</sup><sup>⋆</sup>*HH*′ in **<sup>L</sup>**′ . Furthermore *f*(*a*′ ) is identified to be *f*(**e**−1(*a*′ )).

#### **The double infinitesimal Fourier transform of** exp � <sup>−</sup>*<sup>π</sup>* <sup>⋆</sup>*εξ* <sup>∑</sup>*k*∈*<sup>L</sup> <sup>a</sup>*2(*k*) �

The double infinitesimal Fourier transform of

$$g\_{\xi}(a) = \exp\left(-\pi^{\star}\varepsilon\xi \sum\_{k \in L} a^2(k)\right),\tag{27}$$

where *<sup>ξ</sup>* <sup>∈</sup> **<sup>C</sup>**, Re(*ξ*) <sup>&</sup>gt; 0,

8 ime knjige

and, with *<sup>ε</sup>*<sup>0</sup> := ( <sup>⋆</sup>*H*)<sup>−</sup> ( <sup>⋆</sup> *<sup>H</sup>*)<sup>2</sup>

Then the lattice *L*′

**Theorem 2.2.4.**

quotient space **L**′

/ <sup>∼</sup><sup>⋆</sup>*HH*′ }

)).

to **L**′

be *f*(**e**−1(*a*′

<sup>2</sup> *<sup>H</sup>*′−( <sup>⋆</sup>*H*)<sup>2</sup>

(*F f*)(*b*) := ∑

(*F f*)(*b*) := ∑

(1) *δ* = *F*1 = *F*1, (2) *F* is unitary, *F*<sup>4</sup> = 1, *FF* = *FF* = 1,

The definition directly implies the following proposition :

*<sup>b</sup>*(*k*) = <sup>⋆</sup>( <sup>∗</sup>*β*(*k*)), then st(st(⋆*<sup>ε</sup>* <sup>∑</sup>*k*∈*<sup>L</sup> <sup>a</sup>*(*k*)*b*(*k*))) = <sup>∞</sup>

**Examples of the double infinitesimal Fourier transform**

equivalence class for the equivalence relation <sup>∼</sup><sup>⋆</sup>*HH*′ in **<sup>L</sup>**′

a representative in <sup>⋆</sup>(*L*) satisfying *<sup>k</sup>* <sup>∼</sup>⋆(*H*) <sup>ˆ</sup>

/ <sup>∼</sup><sup>⋆</sup>*HH*′ is defined with *<sup>L</sup>*′

(5) *<sup>F</sup>*(*<sup>f</sup>* <sup>∗</sup> *<sup>g</sup>*)=(*F f*)(*Fg*), (6) *<sup>F</sup>*(*<sup>f</sup>* <sup>∗</sup> *<sup>g</sup>*)=(*F f*)(*Fg*), (7) *<sup>F</sup>*(*f g*)=(*F f*) <sup>∗</sup> (*Fg*), (8) *<sup>F</sup>*(*f g*)=(*F f*) <sup>∗</sup> (*Fg*).

same as the case of the discrete abelian group :

(3) *<sup>f</sup>* <sup>∗</sup> *<sup>δ</sup>* <sup>=</sup> *<sup>δ</sup>* <sup>∗</sup> *<sup>f</sup>* <sup>=</sup> *<sup>f</sup>* , (4) *<sup>f</sup>* <sup>∗</sup> *<sup>g</sup>* <sup>=</sup> *<sup>g</sup>* <sup>∗</sup> *<sup>f</sup>* ,

**Proposition 2.2.5.** If *<sup>l</sup>* <sup>∈</sup> **<sup>R</sup>**+, then

*<sup>a</sup>*∈*<sup>X</sup>*

*<sup>a</sup>*∈*<sup>X</sup>*

*ε*<sup>0</sup> exp

*ε*<sup>0</sup> exp

*Fδ<sup>l</sup>* = (*H*′

The two types of Fourier transforms are different in a standard meaning.

and let **<sup>e</sup>** be a mapping from *<sup>X</sup>* to *XH*, <sup>⋆</sup>*HH*′ , defined by (**e**(*a*))([*k*]) = [*a*(<sup>ˆ</sup>

)(*l*−1)( <sup>⋆</sup>*H*)<sup>2</sup>

If there exists *<sup>α</sup>*, *<sup>β</sup>* <sup>∈</sup> *<sup>L</sup>*2(**R**) so that *<sup>a</sup>* <sup>=</sup> <sup>∗</sup>*α*|*L*, *<sup>b</sup>* <sup>=</sup> <sup>∗</sup>*β*|*L*, that is, *<sup>a</sup>*(*k*) = <sup>⋆</sup>( <sup>∗</sup>*α*(*k*)),

understanding than Definition 2.2.2 for a standard meaning in analysis. For the reason, we consider mainly Definition 2.2.3 about several examples. However Definition 2.2.2 is also treated algebraically, as algebraically defined functions are not always *L*2-functions on **R**.

It is defined: an equivalence relation <sup>∼</sup><sup>⋆</sup>*HH*′ in **<sup>L</sup>**′ by *<sup>x</sup>* <sup>∼</sup><sup>⋆</sup>*HH*′ *<sup>y</sup>* <sup>⇔</sup> *<sup>x</sup>* <sup>−</sup> *<sup>y</sup>* <sup>∈</sup> <sup>⋆</sup>*HH*′ <sup>⋆</sup>( <sup>∗</sup>**Z**). The

. Let *XH*, <sup>⋆</sup>*HH*′ :<sup>=</sup> {*a*′ <sup>|</sup> *<sup>a</sup>*′ is an internal function with a double meaning, from <sup>⋆</sup>**L**/ <sup>∼</sup>⋆(*H*)

left-hand side represents the equivalence class for the equivalence relation <sup>∼</sup>⋆(*H*) in <sup>⋆</sup>**L**, <sup>ˆ</sup>

 <sup>2</sup>*π<sup>i</sup>* <sup>⋆</sup> *<sup>ε</sup>* ∑ *<sup>k</sup>*∈*<sup>L</sup>*

<sup>−</sup>2*π<sup>i</sup>* <sup>⋆</sup>

*<sup>ε</sup>* ∑ *<sup>k</sup>*∈*<sup>L</sup>*

*λµ* is an abelian group for each *λµ*. The following theorem is obtained as

*a*(*k*)*b*(*k*)

*a*(*k*)*b*(*k*)

*f*(*a*) (24)

*f*(*a*). (25)

. (26)

*k*)], where [ ] on the

) is identified to

*k* is

<sup>−</sup><sup>∞</sup> *<sup>α</sup>*(*x*)*b*(*x*)*dx*. Definition 2.2.3 is easier

*k*, and [ ] on the right-hand side represents the

. Furthermore *f*(*a*′

is calculated in the space *<sup>A</sup>* of functionals, for Definition 2.2.3. It is identified <sup>⋆</sup>( <sup>∗</sup>*ξ*) <sup>∈</sup> **<sup>C</sup>** with *<sup>ξ</sup>* <sup>∈</sup> **<sup>C</sup>**.

**Theorem 2.2.6.** *F*(*g<sup>ξ</sup>* )(*b*) = *C<sup>ξ</sup>* (*b*)*g<sup>ξ</sup>* ( *<sup>b</sup> <sup>ξ</sup>* ), where *<sup>b</sup>* <sup>∈</sup> *<sup>X</sup>* and

$$\mathbb{C}\_{\mathfrak{T}}(b) = \sum\_{a \in X} \varepsilon\_0 \exp\left(-\pi^\star \varepsilon\_\xi^\pi \sum\_{k \in L} (a(k) + i\frac{1}{\mathfrak{T}}b(k))^2\right). \tag{28}$$

**Proof.** The infinitesimal Fourier transform of *g<sup>ξ</sup>* (*a*) is done.

$$\begin{aligned} F(g\_{\xi})(b) &= F\left(\exp\left(-\pi^{\star}\varepsilon\xi\sum\_{k\in L}a^2(k)\right)\right)(b) \\ &= \sum\_{a\in X}\varepsilon\_{0}\exp\left(-2i\pi^{\star}\varepsilon\sum\_{k\in L}a(k)b(k)\right)\exp\left(-\pi^{\star}\varepsilon\xi\sum\_{k\in L}a^2(k)\right), \\ &= C\_{\xi}(b)g\_{\xi}(\frac{b}{\xi}). \end{aligned}$$

Let <sup>⋆</sup> ◦ ∗ : **<sup>R</sup>** <sup>→</sup> <sup>⋆</sup>( <sup>∗</sup>**R**) be the natural elementary embedding and let **st**(*c*) for *<sup>c</sup>* <sup>∈</sup> <sup>⋆</sup>( <sup>∗</sup>**R**) be the standard part of *<sup>c</sup>* with respect to the natural elementary embedding <sup>⋆</sup> ◦ ∗. Let st(*c*) be the standard part of *<sup>c</sup>* with respect to the natural elementary embedding <sup>⋆</sup> and <sup>∗</sup> . Then **st** <sup>=</sup> *st* ◦ *st*.

**Theorem 2.2.7.** If the image of *<sup>b</sup>* (<sup>∈</sup> *<sup>X</sup>*) is bounded by a finite value of <sup>∗</sup>**R**, that is, there exists *<sup>b</sup>*<sup>0</sup> <sup>∈</sup> <sup>∗</sup>**<sup>R</sup>** such that *<sup>k</sup>* <sup>∈</sup> *<sup>L</sup>* ⇒ |*b*(*k*)| ≤ <sup>⋆</sup>(*b*0), then

$$\text{st}(\mathbf{C}\_{\tilde{\xi}}(b)) = \left( \* \left( \frac{1}{\sqrt{\xi}} \right) \right)^{H^2} (\in \,^\*\mathbf{R}), \text{ st} \left( \frac{\mathbf{C}\_{\tilde{\xi}}(b)}{\star \left( \left( \* \left( \frac{1}{\sqrt{\xi}} \right) \right)^{H^2} \right)} \right) = 1. \tag{29}$$

**Proof.** st(*C<sup>ξ</sup>* (*b*)) = st(∑*a*∈*<sup>X</sup>* <sup>∏</sup>*k*∈*<sup>L</sup>* <sup>√</sup>*εε*′ exp � <sup>−</sup>*πξ*{ √*ε*(*a*(*k*)) + *i* √*ε* <sup>1</sup> *<sup>ξ</sup>* (*b*(*k*))}<sup>2</sup> � ) <sup>=</sup> <sup>∏</sup>*k*∈*<sup>L</sup>* � <sup>∗</sup><sup>∞</sup> <sup>−</sup> <sup>∗</sup><sup>∞</sup> exp � <sup>−</sup>*πξ*{*<sup>x</sup>* <sup>+</sup> *<sup>i</sup>* √*ε* <sup>1</sup> *<sup>ξ</sup>* st2(*b*(*k*))}<sup>2</sup> � *dx* <sup>=</sup> <sup>∏</sup>*k*∈*<sup>L</sup>* � <sup>∗</sup><sup>∞</sup> <sup>−</sup> <sup>∗</sup><sup>∞</sup> exp � −*πξx*2� *dx*.

The argument is same about the infinitesimal Fourier transform of *g*′ *<sup>ξ</sup>* (*a*) = exp(−*πξ* <sup>∑</sup>*k*∈*<sup>L</sup> <sup>a</sup>*2(*k*)), for Definition 2.2.2, as the above.

**Theorem 2.2.8.**

$$F(g\_{\xi}')(b) = B\_{\xi}(b)g\_{\xi}'(\frac{b}{\overline{\xi}}),\tag{30}$$

where *<sup>b</sup>* <sup>∈</sup> *<sup>X</sup>* and

*<sup>B</sup><sup>ξ</sup>* (*b*) = <sup>∑</sup>*a*∈*<sup>X</sup> <sup>ε</sup>*<sup>0</sup> exp � <sup>−</sup>*πξ* <sup>∑</sup>*k*∈*L*(*a*(*k*) + *<sup>i</sup>* <sup>1</sup> *<sup>ξ</sup> b*(*k*))<sup>2</sup> � . Furthermore, if the image of *<sup>b</sup>* (<sup>∈</sup> *<sup>X</sup>*) is bounded by a finite value of <sup>∗</sup>**R**, that is, <sup>∃</sup>*b*<sup>0</sup> <sup>∈</sup> <sup>∗</sup>**<sup>R</sup>** s.t. *<sup>k</sup>* <sup>∈</sup> *<sup>L</sup>* ⇒ |*b*(*k*)| ≤ <sup>⋆</sup>(*b*0) then

$$\text{st}(B\_{\tilde{\xi}}(b)) = \left( \* \left( \frac{1}{\sqrt{\xi}} \right) \right)^{H^2} (\in \,^\*\mathbf{R}), \text{ st} \left( \frac{B\_{\tilde{\xi}}(b)}{\star \left( \left( \* \left( \frac{1}{\sqrt{\xi}} \right) \right)^{H^2} \right)} \right) = 1. \tag{31}$$

**The double infinitesimal Fourier transform of** exp(−*iπ<sup>m</sup>* <sup>⋆</sup>*<sup>ε</sup>* <sup>∑</sup>*k*∈*<sup>L</sup> <sup>a</sup>*2(*k*))

The double infinitesimal Fourier transform of *gim*(*a*) = exp(−*iπ<sup>m</sup>* <sup>⋆</sup>*<sup>ε</sup>* <sup>∑</sup>*k*∈*<sup>L</sup> <sup>a</sup>*2(*k*)), where *<sup>m</sup>* <sup>∈</sup> **Z**, is calculated for Definition 2.2.3.

**Proposition 2.2.9.** *F*(*gim*)(*b*) is written as *Cim*(*b*)*g* <sup>1</sup> *im* (*b*).

If *<sup>m</sup>*|<sup>2</sup> <sup>⋆</sup>*HH*′<sup>2</sup> and *<sup>m</sup>*<sup>|</sup> *b*(*k*) *<sup>ε</sup>*′ for an arbitrary *<sup>k</sup>* in *<sup>L</sup>*, then *<sup>F</sup>*(*gim*)(*b*) = *Cim*(*b*)*<sup>g</sup>* <sup>1</sup> *im* (*b*), where *Cim*(*b*) = ��*<sup>m</sup>* 2 1+*i* <sup>2</sup> <sup>⋆</sup> *HH*′<sup>2</sup> *m* 1+*i* �( <sup>⋆</sup>*H*)<sup>2</sup> for a positive *m* and *Cim*(*b*) = ��−*<sup>m</sup>* 2 <sup>1</sup>+(−*i*) <sup>2</sup> <sup>⋆</sup> *HH*′<sup>2</sup> −*m* 1−*i* �( <sup>⋆</sup>*H*)<sup>2</sup> for a negative *m*.

**Proof.**

$$F(g\_{im})(b) = \mathbb{C}\_{im}(b)g\_{\frac{1}{m}}(b), \text{ where } \mathbb{C}\_{im}(b) = \sum\_{a \in X} e\_0 \exp\left(-i\pi m' \varepsilon \sum\_{k \in L} (a(k) + \frac{1}{m}b(k))^2\right).$$
 When  $a(k)$ ,  $b(k)$  are denoted as  $\varepsilon' n', \varepsilon' l',$ 

∑<sup>−</sup> <sup>⋆</sup>*<sup>H</sup> <sup>H</sup>*′<sup>2</sup> <sup>2</sup> <sup>≤</sup>*a*(*k*)<sup>&</sup>lt; <sup>⋆</sup>*<sup>H</sup> <sup>H</sup>*′<sup>2</sup> 2 exp(−*iπ<sup>m</sup>* <sup>⋆</sup>*<sup>ε</sup>* <sup>∑</sup>*k*∈*L*(*a*(*k*) + <sup>1</sup> *<sup>m</sup> <sup>b</sup>*(*k*))<sup>2</sup>

$$=\sum\_{-\,^\*H\frac{H'^2}{2}\leq\varepsilon'n'<\,^\*H\frac{H'^2}{2}}\exp(-i\pi m^\star\varepsilon\sum\_{k\in L}(\varepsilon'n'+\varepsilon'\frac{n'}{m})^2).\tag{32}$$

Since *<sup>m</sup>*<sup>|</sup> *b*(*k*) *<sup>ε</sup>*′ , for a positive *m*, it is equal to

$$\sum\_{\substack{\varepsilon \to H \frac{H'^2}{2} \le \varepsilon' \le \varepsilon^\* H \frac{H'^2}{2}}} \exp\left(-i\pi m^\* \varepsilon \varepsilon'^2 n'^2\right) = \frac{m}{2} \sqrt{\frac{2^\* H H'^2}{m}} \frac{1 + i^{\frac{2^\* H H'^2}{m}}}{1 + i} \tag{33}$$

by Proposition 2.1.5. Hence *Cim* = *<sup>m</sup>* 2 1+*i* <sup>2</sup> <sup>⋆</sup> *HH*′<sup>2</sup> *m* 1+*i* ( <sup>⋆</sup>*H*)<sup>2</sup> for a positive *m*. For a negative *m*, the proof is as same as the above.

The argument for the infinitesimal Fourier transform of *g*′ *im*(*a*) = exp(−*iπ<sup>m</sup>* <sup>∑</sup>*k*∈*<sup>L</sup> <sup>a</sup>*2(*k*)), for Definition 2.2.2, is as same as the above one of *gim* for Definition 2.2.3.

**Proposition 2.2.10.** If *<sup>m</sup>*|<sup>2</sup> <sup>⋆</sup>*HH*′<sup>2</sup> and *<sup>m</sup>*<sup>|</sup> *b*(*k*) *<sup>ε</sup>*′ for an arbitrary *<sup>k</sup>* in *<sup>L</sup>*, then (*F*((*g*′ *im*))(*b*) = *Bim*(*b*)*g*′ 1 *im* (*b*), where *Bim*(*b*) = *<sup>m</sup>* 2 1+*i* <sup>2</sup>*H*′<sup>2</sup> *m* 1+*i* ( <sup>⋆</sup>*H*)<sup>2</sup> for a positive *m* and *Bim*(*b*) = <sup>−</sup>*<sup>m</sup>* 2 <sup>1</sup>+(−*i*) <sup>2</sup>*H*′<sup>2</sup> −*m* 1−*i* ( <sup>⋆</sup>*H*)<sup>2</sup> for a negative *m*.

## **2.3. The meaning of the double infinitesimal Fourier transform**

There exists a natural injection from a space of standard functions to *X* as

$$\mathfrak{a} \mapsto (\mathfrak{a} : k \in L \mapsto \star(\,^\*\mathfrak{a}(k)) \in L^\prime). \tag{34}$$

Hence a space of standard functions is embedded in *X* through the natural injection. If there is no confusion, standard functions are identified as nonstandard functions by the natural injection.

For a standard functional *<sup>f</sup>* , if the domain of <sup>⋆</sup>( <sup>∗</sup> *<sup>f</sup>*) is in *<sup>X</sup>*, we can define a Fourier transform *<sup>F</sup>*( <sup>⋆</sup>( <sup>∗</sup> *<sup>f</sup>*)). Since st(st(*F*( <sup>⋆</sup>( <sup>∗</sup> *<sup>f</sup>*))) is a standard functional as st(st(*F*( <sup>⋆</sup>( <sup>∗</sup> *<sup>f</sup>*)))(*α*) =st(st(*F*( <sup>⋆</sup>( <sup>∗</sup> *<sup>f</sup>*))(*a*))) for *<sup>a</sup>* : *<sup>k</sup>* <sup>∈</sup> *<sup>L</sup>* <sup>→</sup> <sup>⋆</sup>( <sup>∗</sup>*α*(*k*)) <sup>∈</sup> *<sup>L</sup>*′ , such standard functional has a Fourier transform st(st(*F*( <sup>⋆</sup>( <sup>∗</sup> *<sup>f</sup>*))).

Similarly to the case of functions, the following subspace <sup>L</sup>2(*A*) of *<sup>A</sup>* is defined:

#### **Definition 2.3.1.**

10 ime knjige

**Theorem 2.2.8.**

where *<sup>b</sup>* <sup>∈</sup> *<sup>X</sup>* and *<sup>B</sup><sup>ξ</sup>* (*b*) = <sup>∑</sup>*a*∈*<sup>X</sup> <sup>ε</sup>*<sup>0</sup> exp

�

� ∗ � 1 √*ξ*

**Proposition 2.2.9.** *F*(*gim*)(*b*) is written as *Cim*(*b*)*g* <sup>1</sup>

�( <sup>⋆</sup>*H*)<sup>2</sup>

�( <sup>⋆</sup>*H*)<sup>2</sup>

*n*′ , *ε*′ *l* ′ ,

exp(−*iπ<sup>m</sup>* <sup>⋆</sup>*<sup>ε</sup>* <sup>∑</sup>*k*∈*L*(*a*(*k*) + <sup>1</sup>

<sup>2</sup> <sup>≤</sup>*ε*′*n*′<sup>&</sup>lt; <sup>⋆</sup>*<sup>H</sup> <sup>H</sup>*′<sup>2</sup>

2

exp(−*iπ<sup>m</sup>* <sup>⋆</sup>*εε*′2*n*′2) = *<sup>m</sup>*

<sup>2</sup> <sup>⋆</sup> *HH*′<sup>2</sup> −*m* 1−*i*

= ∑ <sup>−</sup> <sup>⋆</sup>*<sup>H</sup> <sup>H</sup>*′<sup>2</sup>

*<sup>ε</sup>*′ , for a positive *m*, it is equal to

∑

<sup>2</sup> <sup>≤</sup>*ε*′*n*′<sup>&</sup>lt; <sup>⋆</sup>*<sup>H</sup> <sup>H</sup>*′<sup>2</sup>

2

<sup>−</sup> <sup>⋆</sup>*<sup>H</sup> <sup>H</sup>*′<sup>2</sup>

*b*(*k*)

<sup>1</sup>+(−*i*)

*im*

st(*Bξ* (*b*)) =

**Z**, is calculated for Definition 2.2.3.

If *<sup>m</sup>*|<sup>2</sup> <sup>⋆</sup>*HH*′<sup>2</sup> and *<sup>m</sup>*<sup>|</sup>

��*<sup>m</sup>* 2 1+*i* <sup>2</sup> <sup>⋆</sup> *HH*′<sup>2</sup> *m* 1+*i*

��−*<sup>m</sup>* 2

*F*(*gim*)(*b*) = *Cim*(*b*)*g* <sup>1</sup>

<sup>2</sup> <sup>≤</sup>*a*(*k*)<sup>&</sup>lt; <sup>⋆</sup>*<sup>H</sup> <sup>H</sup>*′<sup>2</sup>

When *a*(*k*), *b*(*k*) are denoted as *ε*′

2

*Cim*(*b*) =

*Cim*(*b*) =

**Proof.**

∑<sup>−</sup> <sup>⋆</sup>*<sup>H</sup> <sup>H</sup>*′<sup>2</sup>

Since *<sup>m</sup>*<sup>|</sup>

*b*(*k*)

*F*(*g*′

��*H*<sup>2</sup>

**The double infinitesimal Fourier transform of** exp(−*iπ<sup>m</sup>* <sup>⋆</sup>*<sup>ε</sup>* <sup>∑</sup>*k*∈*<sup>L</sup> <sup>a</sup>*2(*k*))

<sup>−</sup>*πξ* <sup>∑</sup>*k*∈*L*(*a*(*k*) + *<sup>i</sup>* <sup>1</sup>

*<sup>ξ</sup>* )(*b*) = *B<sup>ξ</sup>* (*b*)*g*′

*<sup>ξ</sup> b*(*k*))<sup>2</sup> �

(<sup>∈</sup> <sup>∗</sup>**R**), **st**

The double infinitesimal Fourier transform of *gim*(*a*) = exp(−*iπ<sup>m</sup>* <sup>⋆</sup>*<sup>ε</sup>* <sup>∑</sup>*k*∈*<sup>L</sup> <sup>a</sup>*2(*k*)), where *<sup>m</sup>* <sup>∈</sup>

for a positive *m* and

for a negative *m*.

*im* (*b*).

*<sup>ε</sup>*′ for an arbitrary *<sup>k</sup>* in *<sup>L</sup>*, then *<sup>F</sup>*(*gim*)(*b*) = *Cim*(*b*)*<sup>g</sup>* <sup>1</sup>

(*b*), where *Cim*(*b*) = <sup>∑</sup>*a*∈*<sup>X</sup> <sup>ε</sup>*<sup>0</sup> exp(−*iπ<sup>m</sup>* <sup>⋆</sup>*<sup>ε</sup>* <sup>∑</sup>*k*∈*L*(*a*(*k*) + <sup>1</sup>

*<sup>m</sup> <sup>b</sup>*(*k*))<sup>2</sup>

exp(−*iπ<sup>m</sup>* <sup>⋆</sup>*<sup>ε</sup>* <sup>∑</sup>

*<sup>k</sup>*∈*<sup>L</sup>* (*ε* ′ *n*′ + *ε* ′ *n*′

2

�<sup>2</sup> <sup>⋆</sup>*HH*′<sup>2</sup> *m*

1 + *i*

<sup>2</sup> <sup>⋆</sup> *HH*′<sup>2</sup> *m*

bounded by a finite value of <sup>∗</sup>**R**, that is, <sup>∃</sup>*b*<sup>0</sup> <sup>∈</sup> <sup>∗</sup>**<sup>R</sup>** s.t. *<sup>k</sup>* <sup>∈</sup> *<sup>L</sup>* ⇒ |*b*(*k*)| ≤ <sup>⋆</sup>(*b*0) then

*ξ* ( *b ξ*

⋆ �� ∗ � √ 1 *ξ*

*Bξ* (*b*)

), (30)

. Furthermore, if the image of *<sup>b</sup>* (<sup>∈</sup> *<sup>X</sup>*) is

<sup>=</sup> 1. (31)

*im*

(*b*), where

*<sup>m</sup> <sup>b</sup>*(*k*))2).

*<sup>m</sup>* )2). (32)

<sup>1</sup> <sup>+</sup> *<sup>i</sup>* (33)

��*H*<sup>2</sup>�

$$\mathcal{L}^2(A) := \{ f \in A \mid \text{there exists } c \in \, ^\*\mathbb{R} \text{ so that } (\frac{1}{c} \sum\_{a \in X} \varepsilon\_0 |f(a)|^2) < +\infty \}. \tag{35}$$

The standard part st(∑*a*∈*<sup>X</sup> <sup>ε</sup>*0<sup>|</sup> *<sup>f</sup>*(*a*)<sup>|</sup> 2) is a ∗− norm in <sup>L</sup>2(*A*). Theorem 2.1.3 (2) implies the following proposition.

**Proposition 2.3.2.** The Fourier transform *<sup>F</sup>* and the inverse *<sup>F</sup>* preserve the space <sup>L</sup>2(*A*).

Hence if *<sup>f</sup>* is a standard functional so that <sup>⋆</sup>( <sup>∗</sup> *<sup>f</sup>*) is an element of <sup>L</sup>2(*A*), the Fourier transformation *<sup>F</sup>*( <sup>⋆</sup>( <sup>∗</sup> *<sup>f</sup>*)), *<sup>F</sup>*( <sup>⋆</sup>( <sup>∗</sup> *<sup>f</sup>*)) are also in <sup>L</sup>2(*A*). Now there is no theory of Fourier transform for functionals in "standard analysis", and it is well-known that there is no nontrivial translation-invariant measure on an infinite-dimensional separable Banach space. In fact, on the infinite-dimensional Banach space there is an infinite sequence of pairwise disjoint open balls of same sizes in a larger ball. The measure is translation-invariant, the measure of the small balls are same, but the measure of the larger ball is finite, it is contradiction. By the reason we do not argue a relationship between our Fourier transform and standard Fourier transform, any more.

Here number fields are extended twice to realize the delta function for functionals. The extended real number field divided to very small infinitesimal lattices. These lattices are too small for normal real number field and the first extended real number field to observe them. Axiomatically, the double extended number field can be treat in a large universe, that is, relative set theory ([13],[14]). The concept of observable and relatively observable are formulated, and two kinds of delta functions are defined. The Fourier transform theory is developed, which is called divergence Fourier transform . It is applied to solve an elementary ordinary differential equation with a delta function(cf.[12]).

## **3. Poisson summation formula**

The Poisson summation formula is a fundamental formula for each Fourier transform theory. In this section, it is explained about the Kinoshita's Fourier transform and our double infinitesimal Fourier transform . Some examples of the gaussian type functions are calculated for the applications of the Poisson summation formula.

## **3.1. Poisson summation formula for infinitesimal Fourier transform**

The Poisson summation formula of finite group is extended to Kinoshita's infinitesimal Fourier transform.

#### **Formulation**

**Theorem 3.1.1.** Let *S* be an internal subgroup of *L*. Then the following formula is obtained, for *<sup>ϕ</sup>* <sup>∈</sup> *<sup>R</sup>*(*L*),

$$\mathbb{E}|\mathcal{S}^{\perp}|^{-\frac{1}{2}}\sum\_{p\in\mathcal{S}^{\perp}}(F\varphi)(p) = |\mathcal{S}|^{-\frac{1}{2}}\sum\_{\mathbf{x}\in\mathcal{S}}\varphi(\mathbf{x})\tag{36}$$

where *<sup>S</sup>*<sup>⊥</sup> :<sup>=</sup> {*<sup>p</sup>* <sup>∈</sup> *<sup>L</sup>* <sup>|</sup> exp(2*πipx*) = 1 for <sup>∀</sup>*<sup>x</sup>* <sup>∈</sup> *<sup>S</sup>*}.

Since *L* is an internal cyclic group, the group *S* is also an internal cyclic group. The generator of *<sup>L</sup>* is *<sup>ε</sup>*. The generator of *<sup>S</sup>* is written as *<sup>ε</sup><sup>s</sup>* (*<sup>s</sup>* <sup>∈</sup> <sup>∗</sup>**Z**). Since the order of *<sup>L</sup>* is *<sup>H</sup>*2, so *<sup>s</sup>* is a factor of *H*2.

The following lemma is prepared for the proof of Theorem 3.1.1.

**Lemma 3.1.2.** *<sup>S</sup>*<sup>⊥</sup> <sup>=</sup><sup>&</sup>lt; *<sup>ε</sup> <sup>H</sup>*<sup>2</sup> *<sup>s</sup>* <sup>&</sup>gt;.

**Proof of Lemma 3.1.2.** For *<sup>p</sup>* <sup>∈</sup> *<sup>S</sup>*⊥, we write *<sup>p</sup>* <sup>=</sup> *<sup>ε</sup>t*. Then the following is obtained:

$$\exp(2\pi i p \text{es}) = 1 \Longleftrightarrow \exp(2\pi i \text{etes}) = 1 \Longleftrightarrow \exp(2\pi i t \frac{s}{H^2}) = 1 \Longleftrightarrow t \frac{s}{H^2} \in \,^\*\mathbf{Z}.\tag{37}$$

Hence the generater of *<sup>S</sup>*<sup>⊥</sup> is *<sup>ε</sup> <sup>H</sup>*<sup>2</sup> *s* .

**Proof of Theorem 3.1.1.** By Lemma 2.1.2, <sup>|</sup>*S*<sup>|</sup> <sup>=</sup> *<sup>H</sup>*<sup>2</sup> *<sup>s</sup>* and <sup>|</sup>*S*⊥| <sup>=</sup> *<sup>s</sup>*. If *<sup>x</sup>* <sup>∈</sup>/ *<sup>S</sup>*, then *<sup>ε</sup> <sup>H</sup>*<sup>2</sup> *<sup>s</sup> xs* = *<sup>ε</sup>H*2*<sup>x</sup>* <sup>∈</sup> <sup>∗</sup>**Z**, and � exp � 2*πiε <sup>H</sup>*<sup>2</sup> *<sup>s</sup> x* ��*s* <sup>=</sup> 1. For *<sup>x</sup>* <sup>∈</sup> *<sup>L</sup>*,

$$\sum\_{p \in S^{\perp}} \exp(2\pi i p \mathbf{x}) = \begin{cases} \frac{\exp\left(2\pi i (-\frac{H}{s})\mathbf{x}\right) \left(1 - \left(\exp\left(2\pi i e \frac{H^2}{s} \mathbf{x}\right)^{\flat}\right)\right)}{1 - \exp\left(2\pi i e \frac{H^2}{s} \mathbf{x}\right)} & \text{( $\mathbf{x} \notin S$ )}\\ \sum\_{p \in S^{\perp}} 1 & \text{( $\mathbf{x} \in S$ )}\\ s = \begin{cases} 0 & \text{( $\mathbf{x} \notin S$ )}\\ s & \text{( $\mathbf{x} \in S$ )} \end{cases} \end{cases} \tag{38}$$

## Hence

12 ime knjige

and standard Fourier transform, any more.

**3. Poisson summation formula**

Fourier transform. **Formulation**

for *<sup>ϕ</sup>* <sup>∈</sup> *<sup>R</sup>*(*L*),

factor of *H*2.

**Lemma 3.1.2.** *<sup>S</sup>*<sup>⊥</sup> <sup>=</sup><sup>&</sup>lt; *<sup>ε</sup> <sup>H</sup>*<sup>2</sup>

ordinary differential equation with a delta function(cf.[12]).

for the applications of the Poisson summation formula.

<sup>|</sup>*<sup>S</sup>*⊥| − 1 <sup>2</sup> ∑ *<sup>p</sup>*∈*S*<sup>⊥</sup>

The following lemma is prepared for the proof of Theorem 3.1.1.

exp(2*πipεs*) = <sup>1</sup> ⇐⇒ exp(2*πiεtεs*) = <sup>1</sup> ⇐⇒ exp(2*πit <sup>s</sup>*

where *<sup>S</sup>*<sup>⊥</sup> :<sup>=</sup> {*<sup>p</sup>* <sup>∈</sup> *<sup>L</sup>* <sup>|</sup> exp(2*πipx*) = 1 for <sup>∀</sup>*<sup>x</sup>* <sup>∈</sup> *<sup>S</sup>*}.

*<sup>s</sup>* <sup>&</sup>gt;.

the measure of the small balls are same, but the measure of the larger ball is finite, it is contradiction. By the reason we do not argue a relationship between our Fourier transform

Here number fields are extended twice to realize the delta function for functionals. The extended real number field divided to very small infinitesimal lattices. These lattices are too small for normal real number field and the first extended real number field to observe them. Axiomatically, the double extended number field can be treat in a large universe, that is, relative set theory ([13],[14]). The concept of observable and relatively observable are formulated, and two kinds of delta functions are defined. The Fourier transform theory is developed, which is called divergence Fourier transform . It is applied to solve an elementary

The Poisson summation formula is a fundamental formula for each Fourier transform theory. In this section, it is explained about the Kinoshita's Fourier transform and our double infinitesimal Fourier transform . Some examples of the gaussian type functions are calculated

The Poisson summation formula of finite group is extended to Kinoshita's infinitesimal

**Theorem 3.1.1.** Let *S* be an internal subgroup of *L*. Then the following formula is obtained,

(*Fϕ*)(*p*) = <sup>|</sup>*S*<sup>|</sup>

Since *L* is an internal cyclic group, the group *S* is also an internal cyclic group. The generator of *<sup>L</sup>* is *<sup>ε</sup>*. The generator of *<sup>S</sup>* is written as *<sup>ε</sup><sup>s</sup>* (*<sup>s</sup>* <sup>∈</sup> <sup>∗</sup>**Z**). Since the order of *<sup>L</sup>* is *<sup>H</sup>*2, so *<sup>s</sup>* is a

**Proof of Lemma 3.1.2.** For *<sup>p</sup>* <sup>∈</sup> *<sup>S</sup>*⊥, we write *<sup>p</sup>* <sup>=</sup> *<sup>ε</sup>t*. Then the following is obtained:

− 1 <sup>2</sup> ∑ *<sup>x</sup>*∈*<sup>S</sup>*

*ϕ*(*x*) (36)

*s*

*<sup>H</sup>*<sup>2</sup> <sup>∈</sup> <sup>∗</sup>**Z**. (37)

*<sup>H</sup>*<sup>2</sup> ) = <sup>1</sup> ⇐⇒ *<sup>t</sup>*

**3.1. Poisson summation formula for infinitesimal Fourier transform**

<sup>∑</sup>*p*∈*S*<sup>⊥</sup> (*Fϕ*)(*p*) = <sup>∑</sup>*p*∈*S*<sup>⊥</sup> *<sup>ε</sup>*(∑*x*∈*<sup>L</sup> <sup>ϕ</sup>*(*x*) exp(2*πipx*))

$$=\varepsilon \sum\_{\mathbf{x}\in L} \varrho(\mathbf{x}) (\sum\_{p\in \mathcal{S}^\perp} \exp(2\pi i p \mathbf{x})) = \frac{\mathbf{s}}{H} \sum\_{\mathbf{x}\in \mathcal{S}} \varrho(\mathbf{x}).\tag{39}$$

Thus

$$\frac{1}{\sqrt{s}} \sum\_{p \in \mathcal{S}^\perp} (F\rho)(p) = \sqrt{\frac{s}{H^2}} \sum\_{\mathbf{x} \in \mathcal{S}} \rho(\mathbf{x}) \tag{40}$$

hence

$$\|S^{\perp}\|^{ - \frac{1}{2}} \sum\_{p \in \mathcal{S}^{\perp}} (F\varphi)(p) = \frac{1}{|\mathcal{S}|^{\frac{1}{2}}} \sum\_{\mathbf{x} \in \mathcal{S}} \varphi(\mathbf{x}) \cdot \cdots \cdot (\sharp\_1). \tag{41}$$

**Proposition 3.1.3.** Especially if *<sup>s</sup>* is equal to *<sup>H</sup>*, then (♯1) implies that <sup>∑</sup>*p*∈*S*<sup>⊥</sup> (*Fϕ*)(*p*) = <sup>∑</sup>*x*∈*<sup>S</sup> <sup>ϕ</sup>*(*x*). The standard part of it is st(∑*p*∈*S*<sup>⊥</sup> (*Fϕ*)(*p*)) =st(∑*x*∈*<sup>S</sup> <sup>ϕ</sup>*(*x*)).

If there exists a standard function *<sup>ϕ</sup>*′ : **<sup>R</sup>** <sup>→</sup> **<sup>C</sup>** so that *<sup>ϕ</sup>* <sup>=</sup> <sup>∗</sup>*ϕ*′ <sup>|</sup>*L*, then the right hand side is equal to <sup>∑</sup>−∞<*x*<<sup>∞</sup> *<sup>ϕ</sup>*′ (*x*), that is, <sup>∑</sup>−∞<*x*<∞st(*ϕ*)(*x*). Furthermore if *<sup>ε</sup><sup>s</sup>* is infinitesimal and *ϕ*′ is integrable on **R**, then

$$\begin{aligned} \text{st}(\varepsilon \sum\_{x \in S} \varrho(x)) &= \int\_{-\infty}^{\infty} \varrho'(x) dx. \\ \text{Since } (\sharp\_1) \text{ implies that} \\ \sum\_{p \in S^\perp} (F\varrho)(p) &= \varepsilon \mathfrak{s} \sum\_{x \in S} \varrho(x), \\ \text{it is obtained } \text{st}(\sum\_{p \in S^\perp} (F\varrho)(p)) &= \int\_{-\infty}^{\infty} \varrho'(x) dx, \text{ that is, } \int\_{-\infty}^{\infty} \mathfrak{s} \mathfrak{t}(\varrho)(x) dx. \end{aligned}$$

The even number *H* is decomposed to prime factors *H* = *pl*<sup>1</sup> <sup>1</sup> *<sup>p</sup>l*<sup>2</sup> <sup>2</sup> ··· *<sup>p</sup>lm <sup>m</sup>* , where *<sup>p</sup>*<sup>1</sup> = 2, *<sup>p</sup>*<sup>1</sup> < *<sup>p</sup>*<sup>2</sup> <sup>&</sup>lt; ··· <sup>&</sup>lt; *pm*, each *pi* is a prime number, 0 <sup>&</sup>lt; *li*. Since *<sup>S</sup>* is a subgroup of *<sup>L</sup>*, the number *s* is a factor of *H*2. When we write *s* as *pk*<sup>1</sup> <sup>1</sup> *<sup>p</sup>k*<sup>2</sup> <sup>2</sup> ··· *<sup>p</sup>km <sup>m</sup>* , the order of *S* is equal to *p*2*l*<sup>1</sup>−*k*<sup>1</sup> <sup>1</sup> *<sup>p</sup>*2*l*<sup>2</sup>−*k*<sup>2</sup> <sup>2</sup> ··· *<sup>p</sup>*2*lm*−*km <sup>m</sup>* and the order of *<sup>S</sup>*<sup>⊥</sup> is *<sup>p</sup>k*<sup>1</sup> <sup>1</sup> *<sup>p</sup>k*<sup>2</sup> <sup>2</sup> ··· *<sup>p</sup>km <sup>m</sup>* . Hence (27) is

$$(p\_1^{k\_1} p\_2^{k\_2} \cdots p\_m^{k\_m})^{-\frac{1}{2}} \sum\_{p \in \mathbb{S}^\perp} (F\varphi)(p)) = (p\_1^{2l\_1 - k\_1} p\_2^{2l\_2 - k\_2} \cdots p\_m^{2l\_m - k\_m})^{-\frac{1}{2}} \sum\_{\mathbf{x} \in \mathbb{S}} \varphi(\mathbf{x}).\tag{42}$$

#### **Examples**

Theorem 3.1.1 is applied to the following two kinds of functions :

$$1.\varphi\_i(\mathbf{x}) = \exp(-i\pi\mathbf{x}^2) \tag{43}$$

$$2\,\varphi\_{\mathfrak{F}}(\mathfrak{x}) = \exp(-\mathfrak{F}\pi\mathfrak{x}^{2})\tag{44}$$

where *<sup>ξ</sup>* <sup>∈</sup> **<sup>C</sup>**, Re(*ξ*) <sup>&</sup>gt; 0. Then the infinitesimal Fourier transforms are :

$$1.(F\varphi\_i)(p) = \exp(-i\frac{\pi}{4})\overline{\varphi\_i(p)}\cdots(\sharp\_2) \tag{45}$$

$$\text{2.} (F\varphi\_{\tilde{\xi}})(p) = c\_{\tilde{\xi}}(p)\varphi\_{\tilde{\xi}}(\frac{p}{\tilde{\xi}}),\tag{46}$$

where st(*c<sup>ξ</sup>* (*p*)) = <sup>√</sup> 1 *<sup>ξ</sup>* , if *p* is finite. Hence the following formulas are obtained :

$$1. |S^\perp|^{-\frac{1}{2}} \exp(-i\frac{\pi}{4}) \sum\_{p \in S^\perp} \overline{\varphi\_i(p)} = |S|^{-\frac{1}{2}} \sum\_{\mathbf{x} \in S} \varphi\_i(\mathbf{x}),\tag{47}$$

$$2\left|S^{\perp}\right|^{-\frac{1}{2}}\sum\_{p\in S^{\perp}}c\_{\tilde{\xi}}(p)\varrho\_{\tilde{\xi}}(\frac{p}{\tilde{\xi}}) = |S|^{-\frac{1}{2}}\sum\_{\mathfrak{x}\in S}\varrho\_{\tilde{\xi}}(\mathfrak{x}).\tag{48}$$

When the generator of *S* is *εs*, this is written as the following, explicitly :

$$1. H \exp(-i\frac{\pi}{4}) \sum\_{p \in S^\perp} \exp(i\pi p^2) = s \sum\_{\mathbf{x} \in S} \exp(-i\pi \mathbf{x}^2) \tag{49}$$

$$2.H \sum\_{p \in \mathcal{S}^\perp} c\_{\tilde{\xi}}(p) \exp(-\frac{1}{\tilde{\xi}}\pi p^2) = s \sum\_{\mathbf{x} \in \mathcal{S}} \exp(-\tilde{\xi}\pi \mathbf{x}^2). \tag{50}$$

The following proposition is obtained:

#### **Proposition 3.1.4.**

(i) If *<sup>s</sup>* <sup>=</sup> *<sup>H</sup>*, then the generator of *<sup>S</sup>* is 1 and *<sup>S</sup>* <sup>=</sup> *<sup>S</sup>*<sup>⊥</sup> <sup>=</sup> *<sup>L</sup>* <sup>∩</sup> <sup>∗</sup>**Z**. Hence

$$1. \exp(-i\frac{\pi}{4}) \sum\_{p \in L \cap \, ^\*\mathbf{Z}} \exp(i\pi p^2) = \sum\_{\mathbf{x} \in L \cap \, ^\*\mathbf{Z}} \exp(-i\pi \mathbf{x}^2) \tag{51}$$

$$2. \sum\_{p \in L \cap \, \Upsilon \, \mathbf{Z}} c\_{\tilde{\xi}}(p) \exp(-\frac{1}{\tilde{\xi}}\pi p^2) = \sum\_{\mathbf{x} \in L \cap \, \Upsilon \, \mathbf{Z}} \exp(-\xi \pi \mathbf{x}^2). \tag{52}$$

Taking their standard parts, we obtain :

14 ime knjige

*p*2*l*<sup>1</sup>−*k*<sup>1</sup> <sup>1</sup> *<sup>p</sup>*2*l*<sup>2</sup>−*k*<sup>2</sup>

**Examples**

(*pk*<sup>1</sup> <sup>1</sup> *<sup>p</sup>k*<sup>2</sup>

where st(*c<sup>ξ</sup>* (*p*)) = <sup>√</sup>

1

1.|*<sup>S</sup>*⊥| − 1

> 2.|*<sup>S</sup>*⊥| − 1 <sup>2</sup> ∑ *<sup>p</sup>*∈*S*<sup>⊥</sup>

1.*<sup>H</sup>* exp(−*<sup>i</sup>*

2.*H* ∑ *<sup>p</sup>*∈*S*<sup>⊥</sup>

The following proposition is obtained:

**Proposition 3.1.4.**

<sup>2</sup> ··· *<sup>p</sup>km*

The even number *H* is decomposed to prime factors *H* = *pl*<sup>1</sup>

number *s* is a factor of *H*2. When we write *s* as *pk*<sup>1</sup>

*<sup>m</sup>* )<sup>−</sup> <sup>1</sup> <sup>2</sup> ∑ *<sup>p</sup>*∈*S*<sup>⊥</sup>

<sup>2</sup> ··· *<sup>p</sup>*2*lm*−*km <sup>m</sup>* and the order of *<sup>S</sup>*<sup>⊥</sup> is *<sup>p</sup>k*<sup>1</sup>

Theorem 3.1.1 is applied to the following two kinds of functions :

where *<sup>ξ</sup>* <sup>∈</sup> **<sup>C</sup>**, Re(*ξ*) <sup>&</sup>gt; 0. Then the infinitesimal Fourier transforms are :

<sup>2</sup> exp(−*<sup>i</sup>*

When the generator of *S* is *εs*, this is written as the following, explicitly :

*<sup>c</sup><sup>ξ</sup>* (*p*)exp(−<sup>1</sup>

(i) If *<sup>s</sup>* <sup>=</sup> *<sup>H</sup>*, then the generator of *<sup>S</sup>* is 1 and *<sup>S</sup>* <sup>=</sup> *<sup>S</sup>*<sup>⊥</sup> <sup>=</sup> *<sup>L</sup>* <sup>∩</sup> <sup>∗</sup>**Z**. Hence

*ξ*

*π* <sup>4</sup> ) ∑ *<sup>p</sup>*∈*S*<sup>⊥</sup>

1.(*Fϕi*)(*p*) = exp(−*<sup>i</sup>*

2.(*Fϕξ* )(*p*) = *c<sup>ξ</sup>* (*p*)*ϕξ* (

*π* <sup>4</sup> ) ∑ *<sup>p</sup>*∈*S*<sup>⊥</sup>

*c<sup>ξ</sup>* (*p*)*ϕξ* (

<sup>1</sup> *<sup>p</sup>l*<sup>2</sup> <sup>2</sup> ··· *<sup>p</sup>lm*

<sup>2</sup> ··· *<sup>p</sup>*2*lm*−*km <sup>m</sup>* )<sup>−</sup> <sup>1</sup>

1.*ϕi*(*x*) = exp(−*iπx*2) (43) 2.*ϕξ* (*x*) = exp(−*ξπx*2) (44)

<sup>4</sup> )*ϕi*(*p*)···(♯2) (45)

), (46)

*ϕi*(*x*), (47)

*ϕξ* (*x*). (48)

exp(−*iπx*2) (49)

exp(−*ξπx*2). (50)

*<sup>m</sup>* . Hence (27) is

<sup>2</sup> ··· *<sup>p</sup>km*

<sup>1</sup> *<sup>p</sup>k*<sup>2</sup>

<sup>2</sup> ··· *<sup>p</sup>km*

<sup>1</sup> *<sup>p</sup>k*<sup>2</sup>

<sup>1</sup> *<sup>p</sup>*2*l*<sup>2</sup>−*k*<sup>2</sup>

*π*

*p ξ*

*<sup>ξ</sup>* , if *p* is finite. Hence the following formulas are obtained :

*<sup>ϕ</sup>i*(*p*) = <sup>|</sup>*S*<sup>|</sup>

*p ξ* ) = <sup>|</sup>*S*<sup>|</sup>

exp(*iπp*2) = *<sup>s</sup>* ∑

*<sup>π</sup>p*2) = *<sup>s</sup>* ∑

− 1 <sup>2</sup> ∑ *<sup>x</sup>*∈*<sup>S</sup>*

− 1 <sup>2</sup> ∑ *<sup>x</sup>*∈*<sup>S</sup>*

*<sup>x</sup>*∈*<sup>S</sup>*

*<sup>x</sup>*∈*<sup>S</sup>*

*<sup>p</sup>*<sup>2</sup> <sup>&</sup>lt; ··· <sup>&</sup>lt; *pm*, each *pi* is a prime number, 0 <sup>&</sup>lt; *li*. Since *<sup>S</sup>* is a subgroup of *<sup>L</sup>*, the

(*Fϕ*)(*p*)) = (*p*2*l*<sup>1</sup>−*k*<sup>1</sup>

*<sup>m</sup>* , where *<sup>p</sup>*<sup>1</sup> = 2, *<sup>p</sup>*<sup>1</sup> <

*ϕ*(*x*). (42)

*<sup>m</sup>* , the order of *S* is equal to

<sup>2</sup> ∑ *<sup>x</sup>*∈*<sup>S</sup>*

$$\begin{split} 2. \text{st}(\sum\_{p \in L \cap \,^\*\mathbf{Z}} c\_{\xi}(p) \text{exp}(-\frac{1}{\xi}\pi p^2)) &= \text{st}(\sum\_{\mathbf{x} \in L \cap \,^\*\mathbf{Z}} \exp(-\xi\pi \mathbf{x}^2)) \\ &= \sum\_{-\infty < n < \infty} \exp(-\xi\pi n^2) = \theta(i\xi) \end{split} \tag{53}$$

where *<sup>θ</sup>*(*z*) is a *<sup>θ</sup>*-function, defined by *<sup>θ</sup>*(*z*) = <sup>∑</sup>−∞<*n*<<sup>∞</sup> exp(*iπzn*2). (ii) If *εs* is infinitesimal, then the equation:

*<sup>H</sup>* <sup>∑</sup>*p*∈*S*<sup>⊥</sup> *<sup>c</sup><sup>ξ</sup>* (*p*)exp(−<sup>1</sup> *<sup>ξ</sup> <sup>π</sup>p*2) = *<sup>s</sup>* <sup>∑</sup>*x*∈*<sup>S</sup>* exp(−*ξπx*2) implies the following:

$$\begin{split} \text{sst}(\sum\_{p \in S^{\perp}} c\_{\xi}(p) \text{exp}(-\frac{1}{\xi}\pi p^{2})) &= \text{sst}(\text{es}\sum\_{\mathbf{x} \in S} \exp(-\xi\pi \mathbf{x}^{2})) \\ &= \int\_{-\infty}^{\infty} \exp(-\xi\pi \mathbf{x}^{2})d\mathbf{x} = \frac{1}{\sqrt{\xi}}. \end{split} \tag{54}$$

It is known that st(*c<sup>ξ</sup>* (*p*)) = <sup>√</sup> 1 *<sup>ξ</sup>* , and <sup>∑</sup>−∞<*x*<<sup>∞</sup> exp(−*ξπx*2) in the formula 2 of (i) is equal to <sup>√</sup> 1 *<sup>ξ</sup>* <sup>∑</sup>−∞<*p*<<sup>∞</sup> exp(−<sup>1</sup> *<sup>ξ</sup> πp*2) by the standard Poisson summation formula. Hence, by 2 of (i), we obtain st(∑*p*∈*S*<sup>⊥</sup> *<sup>c</sup><sup>ξ</sup>* (*p*)exp(−<sup>1</sup> *<sup>ξ</sup> <sup>π</sup>p*2)) = <sup>∑</sup>−∞<*p*<∞st(*c<sup>ξ</sup>* (*p*)exp(−<sup>1</sup> *<sup>ξ</sup> πp*2)).

The formula (♯2) in 1 for *<sup>ϕ</sup>i*(*x*) is extended to *<sup>ϕ</sup>im*(*x*) = exp(−*imπx*2), for an integer *<sup>m</sup>* so that *<sup>m</sup>*|2*H*<sup>2</sup> . If *<sup>m</sup>*<sup>|</sup> *p ε* , we recall

$$(F\varphi\_{im})(p) = c\_{im}(p)\exp(i\pi\frac{1}{m}p^2)\_{\prime\prime}$$

where *cim*(*p*) = *<sup>m</sup>* 2 1+*i* 2*H*2 *m* <sup>1</sup>+*<sup>i</sup>* for a positive *<sup>m</sup>* and *cim*(*p*) = <sup>−</sup>*<sup>m</sup>* 2 <sup>1</sup>+(−*i*) 2*H*2 −*m* <sup>1</sup>−*<sup>i</sup>* for a negative *<sup>m</sup>*. Hence <sup>|</sup>*S*⊥| − 1 <sup>2</sup> <sup>∑</sup>*p*∈*S*<sup>⊥</sup> *cim*(*p*)*<sup>ϕ</sup>* <sup>1</sup> *im* (*p*) = <sup>|</sup>*S*<sup>|</sup> − 1 <sup>2</sup> <sup>∑</sup>*x*∈*<sup>S</sup> <sup>ϕ</sup>im*(*x*). When the generator *<sup>ε</sup>s*′ of *<sup>S</sup>*<sup>⊥</sup> satifies *<sup>m</sup>*|*s*′ , that is, the generator *<sup>ε</sup><sup>s</sup>* of *<sup>S</sup>* satifies *<sup>m</sup>*<sup>|</sup> *<sup>H</sup>*<sup>2</sup> *<sup>s</sup>* , it reduces to the following:

$$H\sqrt{\frac{m}{2}}\frac{1+i^{\frac{2H^{\perp}}{m}}}{1+i}\sum\_{p\in S^{\perp}}\exp(i\pi\frac{1}{m}p^{2})=s\sum\_{\mathbf{x}\in S}\exp(-im\pi\mathbf{x}^{2})\tag{55}$$

for a positive *m*,

$$H\sqrt{\frac{-m}{2}}\frac{1+(-i)^{\frac{2H^{\perp}}{-m}}}{1-i}\sum\_{p\in S^{\perp}}\exp(i\pi\frac{1}{m}p^{2})=s\sum\_{\mathbf{x}\in S}\exp(-im\pi\mathbf{x}^{2})\tag{56}$$

for a negative *m*.

### **3.2. Poisson summation formula for Definition 2.2.2**

Poisson summation formula of finite group is extended to the double infinitesimal Fourier transform for Definition 2.2.2 on the space of functionals.

#### **Formulation**

**Theorem 3.2.1.** Let *Y* be an internal subgroup of *X*. Then the following is obtained, for *<sup>f</sup>* <sup>∈</sup> *<sup>A</sup>*,

$$\left|Y^{\perp}\right|^{-\frac{1}{2}}\sum\_{b\in Y^{\perp}}(Ff)(b) = \left|Y\right|^{-\frac{1}{2}}\sum\_{a\in Y}f(a)\tag{57}$$

where *<sup>Y</sup>*<sup>⊥</sup> :<sup>=</sup> {*<sup>b</sup>* <sup>∈</sup> *<sup>X</sup>* <sup>|</sup> exp(2*π<sup>i</sup>* <sup>&</sup>lt; *<sup>a</sup>*, *<sup>b</sup>* <sup>&</sup>gt;) = 1 for <sup>∀</sup>*<sup>a</sup>* <sup>∈</sup> *<sup>X</sup>*} and <sup>&</sup>lt; *<sup>a</sup>*, *<sup>b</sup>* <sup>&</sup>gt;:<sup>=</sup> <sup>∑</sup>*k*∈*<sup>L</sup> <sup>a</sup>*(*k*)*b*(*k*). **Lemma 3.2.2.** <sup>|</sup>*Y*⊥| <sup>=</sup> <sup>|</sup>*X*<sup>|</sup> |*Y*| .

**Proof of Lemma 3.2.2.** For *<sup>k</sup>* <sup>∈</sup> *<sup>L</sup>*, we denote *Yk* :<sup>=</sup> {*a*(*k*) <sup>∈</sup> *<sup>L</sup>*′ <sup>|</sup> *<sup>a</sup>* <sup>∈</sup> *<sup>Y</sup>*}.

$$\begin{split} b \in Y^{\perp} &\Longleftrightarrow \forall a \in Y, \ \exp(2\pi i \sum\_{k \in L} a(k)b(k)) = 1 \\ \Longleftrightarrow \forall k \in L, \ b(k) \in Y\_{k}^{\perp} \\ \Longleftrightarrow \forall b: L \to L', \ \forall k \in L, \ b(k) \in Y\_{k}^{\perp}. \\ \text{Hence } |Y^{\perp}| = \prod\_{k \in L} |Y\_{k}^{\perp}|. \ \text{Lemma 3.1.2 implies } |Y\_{k}^{\perp}| = \frac{H'^{2}}{|Y\_{k}|}. \ \text{Thus} \end{split}$$

$$|Y^\perp| = \prod\_{k \in L} \left(\frac{H'^2}{|Y\_k|}\right) = \frac{H'^2 {^\ast H^2}}{\prod\_{k \in L} |Y\_k|} = \frac{|X|}{|Y|}\tag{58}$$

**Proof of Theorem 3.2.1.**

$$|Y^{\perp}|^{-\frac{1}{2}}\sum\_{b\in Y^{\perp}}(Ff)(b) = |Y^{\perp}|^{-\frac{1}{2}}\sum\_{a\in X}\varepsilon\_{0}(\sum\_{b\in Y^{\perp}}\exp(-2\pi i < a, b>))f(a). \tag{59}$$

Since <sup>∑</sup>*b*∈*Y*<sup>⊥</sup> exp(−2*π<sup>i</sup>* <sup>&</sup>lt; *<sup>a</sup>*, *<sup>b</sup>* <sup>&</sup>gt;) = <sup>0</sup> (*<sup>a</sup>* <sup>∈</sup>/ *<sup>Y</sup>*) <sup>|</sup>*Y*⊥| (*<sup>a</sup>* <sup>∈</sup> *<sup>Y</sup>*) , the above is equal to

$$|Y^{\perp}|^{-\frac{1}{2}}\varepsilon\_0|Y^{\perp}|\sum\_{a\in Y}f(a) = |Y^{\perp}|^{\frac{1}{2}}H'^{-\*}{}^{H}\sum\_{a\in Y}f(a) = |Y|^{-\frac{1}{2}}\sum\_{a\in Y}f(a). \tag{60}$$

In the special case where *<sup>f</sup>*(*a*) = <sup>∏</sup>*k*∈*<sup>L</sup> fk*(*a*(*k*)),

$$(Ff)(b) = \sum\_{a \in X} \varepsilon\_0 \exp\left(-2\pi i \sum\_{k \in L} a(k)b(k)\right) \prod\_{k \in L} f\_k(a(k))$$

$$=\prod\_{k\in L} (\sum\_{a(k)\in L'} \varepsilon' \exp(-2\pi i a(k)b(k)) f\_k(a(k)).\tag{61}$$

Namely, the Fourier transform in functional space is the product of those in function space.

#### **Corollary 3.2.3.**

16 ime knjige

for a positive *m*,

for a negative *m*.

**Formulation**

**Lemma 3.2.2.** <sup>|</sup>*Y*⊥| <sup>=</sup> <sup>|</sup>*X*<sup>|</sup>

⇐⇒ ∀*<sup>k</sup>* <sup>∈</sup> *<sup>L</sup>*, *<sup>b</sup>*(*k*) <sup>∈</sup> *<sup>Y</sup>*<sup>⊥</sup>

Hence <sup>|</sup>*Y*⊥| <sup>=</sup> <sup>∏</sup>*k*∈*<sup>L</sup>* <sup>|</sup>*Y*<sup>⊥</sup>

**Proof of Theorem 3.2.1.**

<sup>|</sup>*<sup>Y</sup>*⊥| − 1 <sup>2</sup> ∑ *<sup>b</sup>*∈*Y*<sup>⊥</sup>

⇐⇒ *<sup>b</sup>* : *<sup>L</sup>* <sup>→</sup> *<sup>L</sup>*′

*<sup>f</sup>* <sup>∈</sup> *<sup>A</sup>*,

*H* −*m* 2

<sup>1</sup> + (−*i*)

**3.2. Poisson summation formula for Definition 2.2.2**

transform for Definition 2.2.2 on the space of functionals.

<sup>|</sup>*<sup>Y</sup>*⊥| − 1 <sup>2</sup> ∑ *<sup>b</sup>*∈*Y*<sup>⊥</sup>

**Proof of Lemma 3.2.2.** For *<sup>k</sup>* <sup>∈</sup> *<sup>L</sup>*, we denote *Yk* :<sup>=</sup> {*a*(*k*) <sup>∈</sup> *<sup>L</sup>*′ <sup>|</sup> *<sup>a</sup>* <sup>∈</sup> *<sup>Y</sup>*}.

*k* .

<sup>|</sup>*<sup>Y</sup>*⊥| <sup>=</sup> ∏ *<sup>k</sup>*∈*<sup>L</sup>*

(*F f*)(*b*) = <sup>|</sup>*<sup>Y</sup>*⊥|

*<sup>k</sup>* <sup>|</sup>. Lemma 3.1.2 implies <sup>|</sup>*Y*<sup>⊥</sup>

 *H*′<sup>2</sup> |*Yk*|

> − 1 <sup>2</sup> ∑ *<sup>a</sup>*∈*<sup>X</sup>*


*<sup>b</sup>* <sup>∈</sup> *<sup>Y</sup>*<sup>⊥</sup> ⇐⇒ ∀*<sup>a</sup>* <sup>∈</sup> *<sup>Y</sup>*, exp(2*π<sup>i</sup>* <sup>∑</sup>*k*∈*<sup>L</sup> <sup>a</sup>*(*k*)*b*(*k*)) = <sup>1</sup>

, <sup>∀</sup>*<sup>k</sup>* <sup>∈</sup> *<sup>L</sup>*, *<sup>b</sup>*(*k*) <sup>∈</sup> *<sup>Y</sup>*<sup>⊥</sup>

*k*

2*H*2 −*m* <sup>1</sup> <sup>−</sup> *<sup>i</sup>* <sup>∑</sup>

*<sup>p</sup>*∈*S*<sup>⊥</sup>

exp(*iπ*

Poisson summation formula of finite group is extended to the double infinitesimal Fourier

**Theorem 3.2.1.** Let *Y* be an internal subgroup of *X*. Then the following is obtained, for

(*F f*)(*b*) = <sup>|</sup>*Y*<sup>|</sup>

where *<sup>Y</sup>*<sup>⊥</sup> :<sup>=</sup> {*<sup>b</sup>* <sup>∈</sup> *<sup>X</sup>* <sup>|</sup> exp(2*π<sup>i</sup>* <sup>&</sup>lt; *<sup>a</sup>*, *<sup>b</sup>* <sup>&</sup>gt;) = 1 for <sup>∀</sup>*<sup>a</sup>* <sup>∈</sup> *<sup>X</sup>*} and <sup>&</sup>lt; *<sup>a</sup>*, *<sup>b</sup>* <sup>&</sup>gt;:<sup>=</sup> <sup>∑</sup>*k*∈*<sup>L</sup> <sup>a</sup>*(*k*)*b*(*k*).

− 1 <sup>2</sup> ∑ *<sup>a</sup>*∈*<sup>Y</sup>*

*<sup>k</sup>* <sup>|</sup> <sup>=</sup> *<sup>H</sup>*′<sup>2</sup> |*Yk* |

<sup>∏</sup>*k*∈*<sup>L</sup>* <sup>|</sup>*Yk*<sup>|</sup> <sup>=</sup> <sup>|</sup>*X*<sup>|</sup>

<sup>=</sup> *<sup>H</sup>*′<sup>2</sup> <sup>∗</sup>*H*<sup>2</sup>

*<sup>ε</sup>*0( ∑ *<sup>b</sup>*∈*Y*<sup>⊥</sup> . Thus

1

*<sup>m</sup> <sup>p</sup>*2) = *<sup>s</sup>* ∑

*<sup>x</sup>*∈*<sup>S</sup>*

exp(−*imπx*2) (56)

*f*(*a*) (57)

<sup>|</sup>*Y*<sup>|</sup> (58)

exp(−2*π<sup>i</sup>* <sup>&</sup>lt; *<sup>a</sup>*, *<sup>b</sup>* <sup>&</sup>gt;))*f*(*a*). (59)

(i) If each generator of *Yk* is equal to 1, *<sup>f</sup>* is written as <sup>∏</sup>*k*∈*<sup>L</sup> fk*, *fk* <sup>=</sup> <sup>∗</sup>(st(*fk*))|*L*′ , and <sup>∑</sup>−∞<*n*<∞st(*fk*)(*n*) converges, then

$$\operatorname{st}(\sum\_{b \in Y^\perp} (Ff)(b)) = \prod\_{k \in L} (\sum\_{-\infty < n < \infty} \operatorname{st}(f\_k)(n)).\tag{62}$$

(ii) If each generator of *Yk* is infinitesimal, *<sup>f</sup>* is written as <sup>∏</sup>*k*∈*<sup>L</sup> fk*, *fk* <sup>=</sup> <sup>∗</sup>(st(*fk*))|*L*′ and st(*fk*) is *L*1-integrable on **R**, then

$$st(\sum\_{b \in Y^\perp} (Ff)(b)) = \prod\_{k \in L} \int\_{-\infty < t < \infty} st(f\_k)(t)dt. \tag{63}$$

#### **Examples**

Theorem 3.2.1 is applied to the following two kinds of functionals :

$$1. f\_i(a) = \exp\left(-i\pi \sum\_{k \in L} a(k)^2\right) \tag{64}$$

$$2. f\_{\vec{\xi}}(a) = \exp(-\xi \pi \sum\_{k \in L} a(k)^2),\tag{65}$$

where *<sup>ξ</sup>* <sup>∈</sup> **<sup>C</sup>**, Re(*ξ*) <sup>&</sup>gt; 0.

The infinitesimal Fourier transforms of the functionals are :

$$1.(Ff\_i)(b) = (-1)^{\frac{H}{2}} \overline{f\_i(b)} \cdots (\sharp\_3) \tag{66}$$

$$2.(Ff\_{\tilde{\xi}})(b) = B\_{\tilde{\xi}}(b)f\_{\tilde{\xi}}(\frac{b}{\tilde{\xi}}),\tag{67}$$

hence the followings are obtained :

$$\mathbf{1}. |Y^{\perp}|^{-\frac{1}{2}} (-1)^{\frac{H}{2}} \sum\_{b \in Y^{\perp}} \overline{f\_l(b)} = |Y|^{-\frac{1}{2}} \sum\_{a \in Y} f\_l(a) \tag{68}$$

$$2. |Y^{\perp}|^{-\frac{1}{2}} \sum\_{b \in Y^{\perp}} B\_{\tilde{\xi}}(b) f\_{\tilde{\xi}} \left( \frac{b}{\tilde{\xi}} \right) = |Y|^{-\frac{1}{2}} \sum\_{a \in Y} f\_{\tilde{\xi}}(a). \tag{69}$$

These are written as the following, explicitly :

$$|1,|Y^\perp|^{-\frac{1}{2}}(-1)^{\frac{H}{2}}\sum\_{b\in Y^\perp}\exp(-i\pi\sum\_{k\in L}b(k)^2) = |Y|^{-\frac{1}{2}}\sum\_{a\in Y}\exp(-i\pi\sum\_{k\in L}a(k)^2),\tag{70}$$

$$2\left|Y^{\perp}\right|^{-\frac{1}{2}}\sum\_{b\in Y^{\perp}}B\_{\tilde{\xi}}(b)\exp\left(-\frac{1}{\tilde{\xi}}\pi\sum\_{k\in L}b(k)^{2}\right)=|Y|^{-\frac{1}{2}}\sum\_{a\in Y}\exp\left(-\tilde{\xi}\pi\sum\_{k\in L}a(k)^{2}\right).\tag{71}$$

Corollary 3.2.3 implies the following proposition 3.2.4.

#### **Proposition 3.2.4.**

(i) If each generator of *Yk* is equal to 1, then

$$(1.(-1)^{\frac{H}{2}}st(\sum\_{b\in Y^{\perp}}\exp(-i\pi\prod\_{k\in L}b(k)^{2})) = (\sum\_{-\infty$$

$$2.st(\sum\_{b\in Y^\perp} B\_{\tilde{\xi}}(b) \exp(-\frac{1}{\tilde{\xi}}\pi \sum\_{k\in L} b(k)^2)) = (\sum\_{-\infty < n < \infty} \exp(-\tilde{\xi}\pi n^2))^{H^2} \tag{73}$$

.

 = (*θ*(*iξ*))*H*<sup>2</sup> 

(ii) If each generator of *Yk* is equal to a natural number *mk*, then

$$1. (-1)^{\frac{H}{2}} st(\sum\_{b \in Y^\perp} \exp(-i\pi \prod\_{k \in L} b(k)^2)) = \prod\_{k \in L} (m\_k \sum\_{-\infty < n < \infty} \exp(-i\pi m\_k^2 n^2))\tag{74}$$

$$2.st(\sum\_{b\in Y^\perp} B\_{\tilde{\xi}}(b) \exp(-\frac{1}{\tilde{\xi}}\pi \sum\_{k\in L} b(k)^2)) = \prod\_{k\in L} (m\_k \sum\_{-\infty < n < \infty} \exp(-\tilde{\xi}\pi m\_k^2 n^2))\tag{75}$$

$$\left(=\prod\_{k\in L} (m\_k \theta(im\_k^2 \xi))\right).$$

(iii) If each generator of *Yk* is infinitesimal, then

18 ime knjige

The infinitesimal Fourier transforms of the functionals are :

1.|*<sup>Y</sup>*⊥| − 1 <sup>2</sup> (−1) *H* <sup>2</sup> ∑ *<sup>b</sup>*∈*Y*<sup>⊥</sup>

2.|*<sup>Y</sup>*⊥| − 1 <sup>2</sup> ∑ *<sup>b</sup>*∈*Y*<sup>⊥</sup>

These are written as the following, explicitly :

1.|*<sup>Y</sup>*⊥| − 1 <sup>2</sup> (−1) *H* <sup>2</sup> ∑ *<sup>b</sup>*∈*Y*<sup>⊥</sup>

2.|*<sup>Y</sup>*⊥| − 1 <sup>2</sup> ∑ *<sup>b</sup>*∈*Y*<sup>⊥</sup>

**Proposition 3.2.4.**

hence the followings are obtained :

1.(*F fi*)(*b*)=(−1)

2.(*F fξ* )(*b*) = *Bξ* (*b*)*fξ* (

*Bξ* (*b*)*fξ* (

exp(−*i<sup>π</sup>* ∑

*ξ <sup>π</sup>* ∑ *<sup>k</sup>*∈*<sup>L</sup>*

exp(−*i<sup>π</sup>* ∏

*ξ <sup>π</sup>* ∑ *<sup>k</sup>*∈*<sup>L</sup>*

*<sup>B</sup><sup>ξ</sup>* (*b*) exp(−<sup>1</sup>

(ii) If each generator of *Yk* is equal to a natural number *mk*, then

*<sup>k</sup>*∈*<sup>L</sup>*

= (*θ*(*iξ*))*H*<sup>2</sup>

*<sup>B</sup><sup>ξ</sup>* (*b*) exp(−<sup>1</sup>

Corollary 3.2.3 implies the following proposition 3.2.4.

(i) If each generator of *Yk* is equal to 1, then

1.(−1) *H* <sup>2</sup> *st*( ∑ *<sup>b</sup>*∈*Y*<sup>⊥</sup>

2.*st*( ∑ *<sup>b</sup>*∈*Y*<sup>⊥</sup> *<sup>k</sup>*∈*<sup>L</sup>*

*H*

*b ξ*

*fi*(*b*) = <sup>|</sup>*Y*<sup>|</sup>

*b ξ* ) = <sup>|</sup>*Y*<sup>|</sup>

*<sup>b</sup>*(*k*)2) = <sup>|</sup>*Y*<sup>|</sup>

*<sup>b</sup>*(*k*)2) = <sup>|</sup>*Y*<sup>|</sup>

*<sup>b</sup>*(*k*)2)) = ( ∑

*<sup>b</sup>*(*k*)2)) = ( ∑

 .

<sup>−</sup>∞<*n*<<sup>∞</sup>

<sup>−</sup>∞<*n*<<sup>∞</sup>

− 1 <sup>2</sup> ∑ *<sup>a</sup>*∈*<sup>Y</sup>*

− 1 <sup>2</sup> ∑ *<sup>a</sup>*∈*<sup>Y</sup>*

− 1 <sup>2</sup> ∑ *<sup>a</sup>*∈*<sup>Y</sup>*

− 1 <sup>2</sup> ∑ *<sup>a</sup>*∈*<sup>Y</sup>* exp(−*i<sup>π</sup>* ∑

exp(−*ξπ* ∑

exp(−*iπn*2))*H*<sup>2</sup>

exp(−*ξπn*2))*H*<sup>2</sup>

*<sup>k</sup>*∈*<sup>L</sup>*

*<sup>k</sup>*∈*<sup>L</sup>*

<sup>2</sup> *fi*(*b*)···(♯3) (66)

), (67)

*fi*(*a*) (68)

*fξ* (*a*). (69)

*a*(*k*)2), (70)

*a*(*k*)2). (71)

(72)

(73)

$$2.st(\sum\_{b\in Y^\perp} B\_{\tilde{\xi}}(b) \exp(-\frac{1}{\tilde{\xi}}\pi \sum\_{k\in L} b(k)^2)) = (\int\_{-\infty}^{\infty} \exp(-\xi\pi t^2)dt)^{H^2} \tag{76}$$

$$\left(= \left( \* \left( \frac{1}{\sqrt{\xi}} \right) \right)^{H^2} \right).$$

The above formula (♯3) for *fi*(*a*) is extended to *fim*(*a*) = exp(−*im<sup>π</sup>* <sup>∑</sup>*k*∈*<sup>L</sup> <sup>a</sup>*2(*k*)), for an integer *<sup>m</sup>* so that *<sup>m</sup>*|2*H*′<sup>2</sup> . If *<sup>m</sup>*<sup>|</sup> *b*(*k*) *<sup>ε</sup>*′ , we recall

$$(Ff\_{im})(b) = B\_{im}(b)f\_{\frac{1}{\overline{m}}}(b) \tag{77}$$

where *Bim*(*b*) = ��*<sup>m</sup>* 2 1+*i* <sup>2</sup>*H*′<sup>2</sup> *m* 1+*i* �( <sup>⋆</sup>*H*)<sup>2</sup> for a positive *<sup>m</sup>* , *Bim*(*b*) = ��−*<sup>m</sup>* 2 <sup>1</sup>+(−*i*) <sup>2</sup>*H*′<sup>2</sup> −*m* 1−*i* �( <sup>⋆</sup>*H*)<sup>2</sup> for a negative *m*.

Hence <sup>|</sup>*Y*⊥| − 1 <sup>2</sup> <sup>∑</sup>*b*∈*Y*<sup>⊥</sup> *Bim*(*b*)*<sup>f</sup>* <sup>1</sup> *im* (*b*) = <sup>|</sup>*Y*<sup>|</sup> − 1 <sup>2</sup> <sup>∑</sup>*a*∈*<sup>Y</sup> fim*(*a*). When each generator *<sup>ε</sup>*′ *s*′ *<sup>k</sup>* of *Y*<sup>⊥</sup> *k* satisfies *<sup>m</sup>*|*s*′ *<sup>k</sup>*, that is, each generator *<sup>ε</sup>*′ *sk* of *Yk* satisfies *<sup>m</sup>*<sup>|</sup> *<sup>H</sup>*′<sup>2</sup> *sk* , it reduces to the following :

$$H^{("\ \text{H}")^2} \left( \sqrt{\frac{m}{2}} \frac{1 + i^{\frac{2\tilde{H}^2}{m}}}{1 + i} \right)^{("\ \text{H}")^2} \sum\_{b \in Y^\perp} \exp(i\pi \frac{1}{m} \sum\_{k \in L} b(k)^2) = \prod\_{k \in L} s\_k \sum\_{a \in Y} \exp(-im\pi \sum\_{k \in L} a(k)^2) \langle 78 \rangle$$

for a positive *m*, and

$$\begin{split} H^{\prime (\,^\*H)^2} \left( \sqrt{\frac{-m}{2}} \frac{1 + (-i)^{\frac{2H^2}{-m}}}{1 - i} \right)^{(\,^\*H)^2} \sum\_{b \in Y^\perp} \exp(i\pi \frac{1}{m} \sum\_{k \in L} b(k)^2) \\ &= \prod\_{k \in L} s\_k \sum\_{a \in Y} \exp(-im\pi \sum\_{k \in L} a(k)^2) \end{split} \tag{79}$$

for a negative *m*.

If *sk* <sup>=</sup> *<sup>H</sup>*′ and *<sup>m</sup>*|*H*′ , then

$$\left(\sqrt{\frac{m}{2}}\frac{1+i\frac{2H'^2}{m}}{1+i}\right)^{(\
u t)^2} \sum\_{b\in Y^\perp} \exp(i\pi \frac{1}{m} \sum\_{k\in L} b(k)^2) = \sum\_{a\in Y} \exp(-im\pi \sum\_{k\in L} a(k)^2) \tag{80}$$

for a positive *m*, and

$$\left(\sqrt{\frac{-m}{2}}\frac{1+(-i)^{\frac{2H^2}{-m}}}{1-i}\right)^{(\,^\*H)^2} \sum\_{b\in Y^\perp} \exp(i\pi \frac{1}{m} \sum\_{k\in L} b(k)^2) = \sum\_{a\in Y} \exp(-im\pi \sum\_{k\in L} a(k)^2) \tag{81}$$

for a negative *m*, that is,

$$\mathbb{E}\left(\sqrt{m}\exp(-i\frac{\pi}{4})\right)^{("H")^{2}}\sum\_{b\in Y^{\perp}}\exp(i\pi\frac{1}{m}\sum\_{k\in L}b(k)^{2}) = \sum\_{a\in Y}\exp(-im\pi\sum\_{k\in L}a(k)^{2})\tag{82}$$

for a positive *m*, and

$$\mathbb{E}\left(\sqrt{-m}\exp\left(i\frac{\pi}{4}\right)\right)^{\left(^{\ast}H\right)^{2}}\sum\_{b\in Y^{\perp}}\exp\left(i\pi\frac{1}{m}\sum\_{k\in L}b\left(k\right)^{2}\right) = \sum\_{a\in Y}\exp\left(-im\pi\sum\_{k\in L}a\left(k\right)^{2}\right)\tag{83}$$

for a negative *m*.

## **3.3. Poisson summation formula for Definition 2.2.3**

Poisson summation formula of finite group is extended to the double infinitesimal Fourier transformation for Definition 2.2.3 on the space of functionals originally defined in [8].

#### **Formulation**

The following theorem for Definition 2.2.3 is obtained as the argument in the section 3.2.

**Theorem 3.3.1.** Let *Y* be an internal subgroup of *X*. Then the following is obtained, for *<sup>f</sup>* <sup>∈</sup> *<sup>A</sup>*,

$$|Y^{\perp \varepsilon}|^{-\frac{1}{2}} \sum\_{b \in Y^{\perp \varepsilon}} (Ff)(b) = |Y|^{-\frac{1}{2}} \sum\_{a \in Y} f(a) \tag{84}$$

where <sup>&</sup>lt; *<sup>a</sup>*, *<sup>b</sup>* <sup>&</sup>gt;*ε*:<sup>=</sup> <sup>⋆</sup>*<sup>ε</sup>* <sup>∑</sup>*k*∈*<sup>L</sup> <sup>a</sup>*(*k*)*b*(*k*) and *<sup>Y</sup>*⊥*<sup>ε</sup>* :<sup>=</sup> {*<sup>b</sup>* <sup>∈</sup> *<sup>X</sup>* <sup>|</sup> exp(2*π<sup>i</sup>* <sup>&</sup>lt; *<sup>a</sup>*, *<sup>b</sup>* <sup>&</sup>gt;*ε*) = 1 for <sup>∀</sup>*<sup>a</sup>* <sup>∈</sup> *<sup>Y</sup>*}. **Lemma 3.3.2.** <sup>|</sup>*Y*⊥*ε*<sup>|</sup> <sup>=</sup> <sup>|</sup>*X*<sup>|</sup> |*Y*| . **Proof of Lemma 3.3.2.** For *<sup>k</sup>* <sup>∈</sup> *<sup>L</sup>*, it is denoted *Yk* :<sup>=</sup> {*a*(*k*) <sup>∈</sup> *<sup>L</sup>*′ <sup>|</sup> *<sup>a</sup>* <sup>∈</sup> *<sup>Y</sup>*}. *<sup>b</sup>* <sup>∈</sup> *<sup>Y</sup>*⊥*<sup>ε</sup>* ⇐⇒ ∀*<sup>a</sup>* <sup>∈</sup> *<sup>Y</sup>*, exp(2*π<sup>i</sup>* <sup>⋆</sup>*<sup>ε</sup>* <sup>∑</sup>*k*∈*<sup>L</sup> <sup>a</sup>*(*k*)*b*(*k*)) = <sup>1</sup> ⇐⇒ ∀*<sup>k</sup>* <sup>∈</sup> *<sup>L</sup>*, <sup>⋆</sup>*εb*(*k*) <sup>∈</sup> *<sup>Y</sup>*<sup>⊥</sup> *k* . For *<sup>k</sup>* <sup>∈</sup> *<sup>L</sup>*, generators defined by the following are written as *<sup>m</sup>*, *<sup>n</sup>* :

*Yk* <sup>=</sup><sup>&</sup>lt; *<sup>ε</sup>*′ *<sup>m</sup>* <sup>&</sup>gt;, {*b*(*k*) <sup>∈</sup> *<sup>L</sup>*′ <sup>|</sup> <sup>⋆</sup>*εb*(*k*) <sup>∈</sup> *<sup>Y</sup>*<sup>⊥</sup> *<sup>k</sup>* } <sup>=</sup><sup>&</sup>lt; *<sup>ε</sup>*′ *n* > . Now

> exp(2*π<sup>i</sup>* <sup>⋆</sup>*εε*′ *mε* ′ *<sup>n</sup>*) = <sup>1</sup> ⇐⇒ <sup>⋆</sup>*εε*′ *mε* ′ *n* = 1. (85)

It is written *<sup>Y</sup>*⊥*<sup>ε</sup> <sup>k</sup>* :<sup>=</sup> {*b*(*k*) <sup>∈</sup> *<sup>L</sup>*′ <sup>|</sup> <sup>⋆</sup>*εb*(*k*) <sup>∈</sup> *<sup>Y</sup>*<sup>⊥</sup> *<sup>k</sup>* }. Then <sup>|</sup>*Y*⊥*<sup>ε</sup> <sup>k</sup>* <sup>|</sup> <sup>=</sup> *<sup>m</sup>*. This is equal to <sup>⋆</sup>*HH*′<sup>2</sup> <sup>⋆</sup>*HH*′2/*<sup>m</sup>* <sup>=</sup> |*L*′ | |*Yk* | . Hence

$$|Y^{\perp \varepsilon}| = \prod\_{k \in L} |Y\_k^{\perp \varepsilon}| = \frac{|X|}{|Y|}. \tag{86}$$

**Proof of Theorem 3.3.1.**

20 ime knjige

for a positive *m*, and

for a negative *m*.

If *sk* <sup>=</sup> *<sup>H</sup>*′ and *<sup>m</sup>*|*H*′

 �*m* 2

for a positive *m*, and

for a negative *m*, that is,

for a positive *m*, and

for a negative *m*.

**Formulation**

*<sup>f</sup>* <sup>∈</sup> *<sup>A</sup>*,

�√*<sup>m</sup>* exp(−*<sup>i</sup>*

�√−*<sup>m</sup>* exp(*<sup>i</sup>*

 �−*m* 2

*<sup>H</sup>*′( <sup>⋆</sup>*H*)<sup>2</sup>

 �−*m* 2

, then

 

<sup>2</sup>*H*′<sup>2</sup> −*m*

 

�( <sup>⋆</sup>*H*)<sup>2</sup>

�( <sup>⋆</sup>*H*)<sup>2</sup>

**3.3. Poisson summation formula for Definition 2.2.3**

( <sup>⋆</sup>*H*)<sup>2</sup>

∑ *<sup>b</sup>*∈*Y*<sup>⊥</sup>

∑ *<sup>b</sup>*∈*Y*<sup>⊥</sup>

∑ *<sup>b</sup>*∈*Y*<sup>⊥</sup> exp(*iπ*

exp(*iπ*

exp(*iπ*

1 *<sup>m</sup>* ∑ *<sup>k</sup>*∈*<sup>L</sup>*

1 *<sup>m</sup>* ∑ *<sup>k</sup>*∈*<sup>L</sup>*

Poisson summation formula of finite group is extended to the double infinitesimal Fourier transformation for Definition 2.2.3 on the space of functionals originally defined in [8].

The following theorem for Definition 2.2.3 is obtained as the argument in the section 3.2. **Theorem 3.3.1.** Let *Y* be an internal subgroup of *X*. Then the following is obtained, for

1 *<sup>m</sup>* ∑ *<sup>k</sup>*∈*<sup>L</sup>*

( <sup>⋆</sup>*H*)<sup>2</sup>

∑ *<sup>b</sup>*∈*Y*<sup>⊥</sup> exp(*iπ*

1 + *i* <sup>2</sup>*H*′<sup>2</sup> *m* 1 + *i*

<sup>1</sup> + (−*i*)

<sup>1</sup> <sup>−</sup> *<sup>i</sup>*

*π* 4 )

*π* 4 ) <sup>1</sup> + (−*i*)

<sup>1</sup> <sup>−</sup> *<sup>i</sup>*

<sup>2</sup>*H*′<sup>2</sup> −*m*

 

= ∏ *<sup>k</sup>*∈*<sup>L</sup>*

> 1 *<sup>m</sup>* ∑ *<sup>k</sup>*∈*<sup>L</sup>*

( <sup>⋆</sup>*H*)<sup>2</sup>

*sk* ∑ *<sup>a</sup>*∈*<sup>Y</sup>*

∑ *<sup>b</sup>*∈*Y*<sup>⊥</sup>

*<sup>b</sup>*(*k*)2) = ∑

*<sup>a</sup>*∈*<sup>Y</sup>*

*<sup>b</sup>*(*k*)2) = ∑

*<sup>b</sup>*(*k*)2) = ∑

*<sup>b</sup>*(*k*)2) = ∑

*<sup>a</sup>*∈*<sup>Y</sup>*

*<sup>a</sup>*∈*<sup>Y</sup>*

*<sup>a</sup>*∈*<sup>Y</sup>*

exp(*iπ*

exp(−*im<sup>π</sup>* ∑

1 *<sup>m</sup>* ∑ *<sup>k</sup>*∈*<sup>L</sup>*

*<sup>k</sup>*∈*<sup>L</sup>*

exp(−*im<sup>π</sup>* ∑

exp(−*im<sup>π</sup>* ∑

exp(−*im<sup>π</sup>* ∑

exp(−*im<sup>π</sup>* ∑

*<sup>k</sup>*∈*<sup>L</sup>*

*<sup>k</sup>*∈*<sup>L</sup>*

*<sup>k</sup>*∈*<sup>L</sup>*

*<sup>k</sup>*∈*<sup>L</sup>*

*b*(*k*)2)

*a*(*k*)2) (79)

*a*(*k*)2) (80)

*a*(*k*)2) (81)

*a*(*k*)2) (82)

*a*(*k*)2) (83)

$$|Y^{\perp \varepsilon}|^{ - \frac{1}{2}} \sum\_{b \in Y^{\perp \varepsilon}} (Ff)(b) = |Y^{\perp \varepsilon}|^{ - \frac{1}{2}} \sum\_{a \in X} \varepsilon\_0 (\sum\_{b \in Y^{\perp \varepsilon}} \exp(-2\pi i < a, b > \varepsilon)) f(a). \tag{87}$$

Since <sup>∑</sup>*b*∈*Y*⊥*<sup>ε</sup>* exp(−2*π<sup>i</sup>* <sup>&</sup>lt; *<sup>a</sup>*, *<sup>b</sup>* <sup>&</sup>gt;*ε*) = <sup>0</sup> (*<sup>a</sup>* <sup>∈</sup>/ *<sup>Y</sup>*) <sup>|</sup>*Y*⊥*ε*<sup>|</sup> (*<sup>a</sup>* <sup>∈</sup> *<sup>Y</sup>*) , the above is equal to

$$|Y^{\perp \varepsilon}|^{-\frac{1}{2}} \varepsilon\_0 |Y^{\perp \varepsilon}| \sum\_{a \in Y} f(a) = |Y|^{-\frac{1}{2}} \sum\_{a \in Y} f(a). \tag{88}$$

The following is obtained:

#### **Corollary 3.3.3.**

(i) If each generator of *Yk* is equal to 1, *<sup>f</sup>* is written as <sup>∏</sup>*k*∈*<sup>L</sup> fk*, *fk* <sup>=</sup> <sup>∗</sup>(st(*fk*))|*L*′ , and <sup>∑</sup>−∞<*n*<∞st(*fk*)(*n*) converges, then

$$H^{\frac{H^{\mathbb{L}}{2}}{2}}st(\sum\_{b \in Y^{\perp}} (Ff)(b)) = \prod\_{k \in L} (\sum\_{-\infty < n < \infty} st(f\_k)(n)).\tag{89}$$

(ii) If each generator of *Yk* is infinitesimal, *<sup>f</sup>* is written as <sup>∏</sup>*k*∈*<sup>L</sup> fk*, *fk* <sup>=</sup> <sup>∗</sup>(st(*fk*))|*L*′ , and st(*fk*) is *L*1-integrable on **R**, then

$$H^{\frac{H^2}{2}}st(\sum\_{b\in Y^\perp} (Ff)(b)) = \prod\_{\tilde{k}\in L} \int\_{-\infty}^\infty st(f\_k)(t)dt.\tag{90}$$

**Examples** Theorem 3.3.1 is applied to the following two functionals :

$$1. \mathcal{g}\_i(a) = \exp(-i\pi \, ^\star \varepsilon \sum\_{k \in L} a(k)^2) \tag{91}$$

$$2.g\_{\mathfrak{F}}(a) = \exp(-\mathfrak{F}\pi^{\star}\varepsilon \sum\_{k \in L} a(k)^2) \tag{92}$$

where *<sup>ξ</sup>* <sup>∈</sup> **<sup>C</sup>**, Re(*ξ*) <sup>&</sup>gt; 0. The infinitesimal Fourier transforms are :

$$1.(Fg\_i)(b) = (-1)^{\frac{kl}{2}} \overline{g\_i(b)} \cdots (\sharp\_4) \tag{93}$$

$$\mathcal{Z}.(Fg\_{\tilde{\xi}})(b) = \mathcal{C}\_{\tilde{\xi}}(b)g\_{\tilde{\xi}}(\frac{b}{\tilde{\xi}}) \tag{94}$$

hence the following formulas are obtained :

$$\left|1.\left|Y^{\perp\mathcal{E}}\right|^{-\frac{1}{2}}(-1)^{\frac{H}{2}}\sum\_{b\in Y^{\perp\mathcal{E}}}\overline{g\_i(b)} = \left|Y\right|^{-\frac{1}{2}}\sum\_{a\in Y}g\_i(a)\tag{95}$$

$$2\left|Y^{\perp\varepsilon}\right|^{-\frac{1}{2}}\sum\_{b\in Y^{\perp\varepsilon}}\mathsf{C}\_{\xi}(b)\mathfrak{g}\_{\xi}(\frac{b}{\xi}) = |Y|^{-\frac{1}{2}}\sum\_{a\in Y}\mathsf{g}\_{\xi}(a).\tag{96}$$

These are written as the following, explicitly :

$$|1.|Y^{\perp\varepsilon}|^{-\frac{1}{2}}(-1)^{\frac{H}{2}}\sum\_{b\in Y^{\perp\varepsilon}}\exp(-i\pi^{\star}\varepsilon\sum\_{k\in L}b(k)^{2})=|Y|^{-\frac{1}{2}}\sum\_{a\in Y}\exp(-i\pi^{\star}\varepsilon\sum\_{k\in L}a(k)^{2})\tag{97}$$

$$|2\!/Y^{\perp\varepsilon}|^{-\frac{1}{2}}\sum\_{b\in Y^{\perp\varepsilon}}\mathsf{C}\_{\mathsf{f}}(b)\exp\left(-\frac{1}{\mathsf{f}}\pi^{\star}\varepsilon\sum\_{k\in L}a(k)^{2}\right) = |Y|^{-\frac{1}{2}}\sum\_{a\in Y}\exp\left(-\mathsf{f}\pi^{\star}\varepsilon\sum\_{k\in L}a(k)^{2}\right).\tag{98}$$

Corollaly 3.3.3 implies the following proposition 3.3.4.

#### **Proposition 3.3.4.**

22 ime knjige

**Corollary 3.3.3.**

<sup>∑</sup>−∞<*n*<∞st(*fk*)(*n*) converges, then

st(*fk*) is *L*1-integrable on **R**, then

*<sup>H</sup> <sup>H</sup>*<sup>2</sup>

*<sup>H</sup> <sup>H</sup>*<sup>2</sup>

hence the following formulas are obtained :

1.|*Y*⊥*<sup>ε</sup>* | − 1 <sup>2</sup> (−1) *H* <sup>2</sup> ∑ *<sup>b</sup>*∈*Y*⊥*<sup>ε</sup>*

2.|*Y*⊥*<sup>ε</sup>* | − 1 <sup>2</sup> ∑ *<sup>b</sup>*∈*Y*⊥*<sup>ε</sup>*

*<sup>C</sup><sup>ξ</sup>* (*b*)exp(−<sup>1</sup>

Corollaly 3.3.3 implies the following proposition 3.3.4.

These are written as the following, explicitly :

1.|*Y*⊥*<sup>ε</sup>* | − 1 <sup>2</sup> (−1) *H* <sup>2</sup> ∑ *<sup>b</sup>*∈*Y*⊥*<sup>ε</sup>*

2.|*Y*⊥*<sup>ε</sup>* | − 1 <sup>2</sup> ∑ *<sup>b</sup>*∈*Y*⊥*<sup>ε</sup>* <sup>2</sup> *st*( ∑ *<sup>b</sup>*∈*Y*<sup>⊥</sup>

**Examples** Theorem 3.3.1 is applied to the following two functionals :

where *<sup>ξ</sup>* <sup>∈</sup> **<sup>C</sup>**, Re(*ξ*) <sup>&</sup>gt; 0. The infinitesimal Fourier transforms are :

<sup>2</sup> *st*( ∑ *<sup>b</sup>*∈*Y*<sup>⊥</sup>

(i) If each generator of *Yk* is equal to 1, *<sup>f</sup>* is written as <sup>∏</sup>*k*∈*<sup>L</sup> fk*, *fk* <sup>=</sup> <sup>∗</sup>(st(*fk*))|*L*′ , and

*<sup>k</sup>*∈*<sup>L</sup>*

(ii) If each generator of *Yk* is infinitesimal, *<sup>f</sup>* is written as <sup>∏</sup>*k*∈*<sup>L</sup> fk*, *fk* <sup>=</sup> <sup>∗</sup>(st(*fk*))|*L*′ , and

*<sup>k</sup>*∈*<sup>L</sup>*

*H*

 ∞ −∞

*<sup>k</sup>*∈*<sup>L</sup>*

*<sup>k</sup>*∈*<sup>L</sup>*

*b ξ*

*gi*(*b*) = <sup>|</sup>*Y*<sup>|</sup>

*<sup>b</sup>*(*k*)2) = <sup>|</sup>*Y*<sup>|</sup>

*<sup>a</sup>*(*k*)2) = <sup>|</sup>*Y*<sup>|</sup>

*b ξ* ) = <sup>|</sup>*Y*<sup>|</sup> − 1 <sup>2</sup> ∑ *<sup>a</sup>*∈*<sup>Y</sup>*

− 1 <sup>2</sup> ∑ *<sup>a</sup>*∈*<sup>Y</sup>*

> − 1 <sup>2</sup> ∑ *<sup>a</sup>*∈*<sup>Y</sup>*

− 1 <sup>2</sup> ∑ *<sup>a</sup>*∈*<sup>Y</sup>*

(*F f*)(*b*)) = ∏

1.*gi*(*a*) = exp(−*i<sup>π</sup>* <sup>⋆</sup>*<sup>ε</sup>* <sup>∑</sup>

2.*g<sup>ξ</sup>* (*a*) = exp(−*ξπ* <sup>⋆</sup>*<sup>ε</sup>* <sup>∑</sup>

1.(*Fgi*)(*b*)=(−1)

2.(*Fgξ* )(*b*) = *Cξ* (*b*)*gξ* (

*Cξ* (*b*)*gξ* (

*<sup>k</sup>*∈*<sup>L</sup>*

exp(−*i<sup>π</sup>* <sup>⋆</sup>*<sup>ε</sup>* <sup>∑</sup>

*ξ <sup>π</sup>* <sup>⋆</sup>*<sup>ε</sup>* ∑ *<sup>k</sup>*∈*<sup>L</sup>* ( ∑ <sup>−</sup>∞<*n*<<sup>∞</sup> *st*(*fk*)(*n*)). (89)

*st*(*fk*)(*t*)*dt*. (90)

*a*(*k*)2) (91)

*a*(*k*)2) (92)

<sup>2</sup> *gi*(*b*)···(♯4) (93)

exp(−*i<sup>π</sup>* <sup>⋆</sup>*<sup>ε</sup>* <sup>∑</sup>

exp(−*ξπ* <sup>⋆</sup>*<sup>ε</sup>* <sup>∑</sup>

) (94)

*gi*(*a*) (95)

*gξ* (*a*). (96)

*<sup>k</sup>*∈*<sup>L</sup>*

*<sup>k</sup>*∈*<sup>L</sup>*

*a*(*k*)2) (97)

*a*(*k*)2). (98)

(*F f*)(*b*)) = ∏

(i) If each generator of *Yk* is equal to 1, then the standard parts are :

$$1. H^{\frac{H^2}{2}}(-1)^{\frac{H}{2}} \text{st}(\sum\_{b \in Y\_i^\perp} \exp(-i\pi e \sum\_{k \in L} b(k)^2)) = (\sum\_{-\infty < n < \infty} \exp(-i\pi e n^2))^{H^2} \tag{99}$$

$$2. H^{\frac{H^2}{2}} st(\sum\_{b \in Y\_\ell^\perp} \mathbb{C}\_\xi(b) \exp(-\frac{1}{\xi}\pi\epsilon \sum\_{k \in L} b(k)^2)) = (\sum\_{-\infty < n < \infty} \exp(-\xi\pi\epsilon n^2))^{H^2} \tag{100}$$

 = (*θ*(*iξ*))*H*<sup>2</sup> .

(ii) If each generator of *Yk* is equal to a natural number *mk*, then

$$1.1H^{\frac{H^2}{2}}(-1)^{\frac{H}{2}}st(\sum\_{b\in Y\_{\ell}^\perp} \exp(-i\pi\varepsilon \sum\_{k\in L} b(k)^2)) = \prod\_{k\in L} (m\_k \sum\_{-\infty$$

$$2. H^{\frac{H^2}{2}} st(\sum\_{b \in Y\_\ell^\perp} \mathbb{C}\_\xi(b) \exp(-\frac{1}{\xi}\pi\varepsilon \sum\_{k \in L} b(k)^2)) = \prod\_{k \in L} (m\_k \sum\_{-\infty < n < \infty} \exp(-\xi\pi\varepsilon m\_k^2 n^2))\tag{102}$$

$$\left(=\prod\_{k\in L} (m\_k \theta(im\_k^2 \xi))\right).$$

(iii) If each generator of *Yk* is infinitesimal, then

$$2.st(\sum\_{b\in Y\_{\xi}^{\perp}} \mathbb{C}\_{\xi}(b)\exp(-\frac{1}{\xi}\pi\varepsilon\sum\_{k\in L}b(k)^{2}))=(\int\_{-\infty}^{\infty}\exp(-\xi\pi t^{2})dt)^{H^{2}}\tag{103}$$

$$\left( = \left( \* \left( \frac{1}{\sqrt{\xi}} \right) \right)^{H^2} \right) \cdot$$

The above formulation (♯4) of *gi*(*a*) is extended to *gim*(*a*) = exp(−*im<sup>π</sup>* <sup>⋆</sup>*<sup>ε</sup>* <sup>∑</sup>*k*∈*<sup>L</sup> <sup>a</sup>*2(*k*)), for an integer *<sup>m</sup>* so that *<sup>m</sup>*|<sup>2</sup> <sup>⋆</sup>*HH*′<sup>2</sup> . If *<sup>m</sup>*<sup>|</sup> *b*(*k*) *<sup>ε</sup>*′ for an arbitrary *<sup>k</sup>* <sup>∈</sup> *<sup>L</sup>*, it is recalled

$$\begin{array}{rcl} \mathrm{C}(F\mathrm{g}\_{\mathrm{im}})(b) &=& \mathrm{C}\_{\mathrm{im}}(b)\mathrm{g}\_{\frac{1}{\mathrm{m}}}(b), \text{ where } \mathrm{C}\_{\mathrm{im}}(b) &=& \left(\sqrt{\frac{m}{2}}\frac{1+i^{2}\frac{H^{2}}{1+i}}{1+i}\right)^{\mathrm{\*}H^{2}} \text{ for a positive } m \text{ and} \\ \mathrm{C}\_{\mathrm{im}}(b) &=& \left(\sqrt{\frac{-m}{2}}\frac{1+(-i)^{\frac{2^{\*}H^{2}}{1-i}}}{1-i}\right)^{\mathrm{\*}H^{2}} \text{ for a negative } m. \end{array}$$

Hence <sup>|</sup>*Y*⊥*<sup>ε</sup>* <sup>|</sup> − 1 <sup>2</sup> <sup>∑</sup>*b*∈*Y*<sup>⊥</sup> *Cim*(*b*)*<sup>g</sup>* <sup>1</sup> *im* (*b*) = <sup>|</sup>*Y*<sup>|</sup> − 1 <sup>2</sup> <sup>∑</sup>*a*∈*<sup>Y</sup> gim*(*a*). When each generator *<sup>ε</sup>*′ *s*′ *<sup>k</sup>* of *Y*⊥*<sup>ε</sup> k* satisfies *<sup>m</sup>*|*s*′ *<sup>k</sup>*, that is, each generator *ε*′ *sk* of *Yk* satisfies *<sup>m</sup>*<sup>|</sup> <sup>⋆</sup>*HH*′<sup>2</sup> *sk* , it reduces to the following:

$$\begin{aligned} \, \_H H^{\frac{H^2}{2}} H^{(\ \ \*H)^2} \left( \sqrt{\frac{m}{2}} \frac{1 + i \frac{\star\_{HH^2}}{m}}{1 + i} \right)^{(\ \*H)^2} \sum\_{\substack{b \in Y^{\perp\_\varepsilon} \\ b \in Y^{\perp\_\varepsilon}}} \exp(i\pi \frac{1}{m} \, ^\star \varepsilon \sum\_{k \in L} b(k)^2) \\\\ = \prod\_{k \in L} s\_k \sum\_{a \in Y} \exp(-im\pi \, ^\star \varepsilon \sum\_{k \in L} a(k)^2) \end{aligned} \tag{104}$$

for a positive *m*, and

$$\begin{aligned} \, \_H H^{\frac{H^2}{2}} H^{(\, \*H \,)^2} \left( \sqrt{\frac{-m}{2}} \frac{1 + (-i)^{\frac{2 \ast\_{HH^2}}{-m}}}{1 - i} \right)^{(\, \*H \,)^2} \sum\_{b \in Y^{\perp\_d}} \exp(i \pi \frac{1}{m} \, ^\star \varepsilon \sum\_{k \in L} b(k)^2) \\\\ = \prod\_{k \in L} s\_k \sum\_{a \in Y} \exp(-i m \pi \, ^\star \varepsilon \sum\_{k \in L} a(k)^2) \end{aligned} \tag{105}$$

*<sup>k</sup>*∈*<sup>L</sup>*

for a negative *<sup>m</sup>*. If *sk* <sup>=</sup> *<sup>H</sup>*′ and *<sup>m</sup>*|*H*′ , then

*<sup>k</sup>*∈*<sup>L</sup>*

*<sup>a</sup>*∈*<sup>Y</sup>*

$$H^{\frac{H^2}{2}}\left(\sqrt{\frac{m}{2}}\frac{1+i^{\frac{2^\star + 4H^\star - 2}{m}}}{1+i}\right)^{(\ \*H)^2} \sum\_{b \in Y^{\perp\_d}} \exp(i\pi \frac{1}{m} \sum\_{k \in L} b(k)^2) = \sum\_{a \in Y} \exp(-im\pi \,^\star \varepsilon \sum\_{k \in L} a(k)^2) \tag{106}$$

for a positive *m*, and

$$\begin{split} H^{\frac{H^{2}}{2}} \left( \sqrt{\frac{-m}{2}} \frac{1 + (-i)^{\frac{2^{\star}HH^{2}}{-m}}}{1 - i} \right)^{(\ \star H)^{2}} \sum\_{b \in Y^{\perp\_{d}}} \exp(i\pi \frac{1}{m} \sum\_{k \in L} b(k)^{2}) \\ = \sum\_{a \in Y} \exp(-im\pi \, ^{\star}\varepsilon \sum\_{k \in L} a(k)^{2}) \end{split} \tag{107}$$

for a negative *m*, that is,

$$\frac{1}{2}H^{\frac{H^2}{2}}\left(\sqrt{m}\exp(-i\frac{\pi}{4})\right)^{(\ast^\*H)^2}\sum\_{b\in Y^{\perp\_\varepsilon}}\exp(i\pi\frac{1}{m}\sum\_{k\in L}b(k)^2) = \sum\_{a\in Y}\exp(-im\pi^\*\varepsilon\sum\_{k\in L}a(k)^2) \tag{108}$$

for a positive *m*, and

24 ime knjige

Hence <sup>|</sup>*Y*⊥*<sup>ε</sup>* <sup>|</sup>

satisfies *<sup>m</sup>*|*s*′

− 1

*<sup>H</sup> <sup>H</sup>*<sup>2</sup>

for a positive *m*, and

*<sup>H</sup> <sup>H</sup>*<sup>2</sup> 2 �*m* 2

for a positive *m*, and

for a negative *m*, that is,

�√*<sup>m</sup>* exp(−*<sup>i</sup>*

*π* 4 )

�( <sup>⋆</sup>*H*)<sup>2</sup>

*<sup>H</sup> <sup>H</sup>*<sup>2</sup> 2

*<sup>H</sup> <sup>H</sup>*<sup>2</sup>

<sup>2</sup> *<sup>H</sup>*′( <sup>⋆</sup>*H*)<sup>2</sup>

for a negative *<sup>m</sup>*. If *sk* <sup>=</sup> *<sup>H</sup>*′ and *<sup>m</sup>*|*H*′

<sup>2</sup> <sup>⋆</sup> *HH*′<sup>2</sup> *m* 1 + *i*

  ( <sup>⋆</sup>*H*)<sup>2</sup>

1 + *i*

*<sup>H</sup> <sup>H</sup>*<sup>2</sup> 2 �−*m* 2

 �−*m* 2

<sup>2</sup> <sup>∑</sup>*b*∈*Y*<sup>⊥</sup> *Cim*(*b*)*<sup>g</sup>* <sup>1</sup>

<sup>2</sup> *<sup>H</sup>*′( <sup>⋆</sup>*H*)<sup>2</sup>

*<sup>k</sup>*, that is, each generator *ε*′

 �*m* 2

*im*

= ∏ *<sup>k</sup>*∈*<sup>L</sup>*

= ∏ *<sup>k</sup>*∈*<sup>L</sup>*

(*b*) = <sup>|</sup>*Y*<sup>|</sup>

<sup>2</sup> <sup>⋆</sup> *HH*′<sup>2</sup> *m* 1 + *i*

1 + *i*

*sk* ∑ *<sup>a</sup>*∈*<sup>Y</sup>*

<sup>1</sup> + (−*i*)

*sk* ∑ *<sup>a</sup>*∈*<sup>Y</sup>*

∑ *<sup>b</sup>*∈*Y*⊥*<sup>ε</sup>*

<sup>1</sup> + (−*i*)

∑ *<sup>b</sup>*∈*Y*⊥*<sup>ε</sup>*

<sup>1</sup> <sup>−</sup> *<sup>i</sup>*

, then

exp(*iπ*

<sup>2</sup> <sup>⋆</sup> *HH*′<sup>2</sup> −*m*

exp(*iπ*

1 *<sup>m</sup>* ∑ *<sup>k</sup>*∈*<sup>L</sup>*

  ( <sup>⋆</sup>*H*)<sup>2</sup>

= ∑ *<sup>a</sup>*∈*<sup>Y</sup>*

∑ *<sup>b</sup>*∈*Y*⊥*<sup>ε</sup>*

1 *<sup>m</sup>* ∑ *<sup>k</sup>*∈*<sup>L</sup>*

<sup>1</sup> <sup>−</sup> *<sup>i</sup>*

<sup>2</sup> <sup>⋆</sup> *HH*′<sup>2</sup> −*m*

− 1

 

*sk* of *Yk* satisfies *<sup>m</sup>*<sup>|</sup>

( <sup>⋆</sup>*H*)<sup>2</sup>

exp(−*im<sup>π</sup>* <sup>⋆</sup>*<sup>ε</sup>* <sup>∑</sup>

 

exp(−*im<sup>π</sup>* <sup>⋆</sup>*<sup>ε</sup>* <sup>∑</sup>

( <sup>⋆</sup>*H*)<sup>2</sup>

∑ *<sup>b</sup>*∈*Y*⊥*<sup>ε</sup>*

*<sup>k</sup>*∈*<sup>L</sup>*

∑ *<sup>b</sup>*∈*Y*⊥*<sup>ε</sup>*

*<sup>k</sup>*∈*<sup>L</sup>*

*<sup>b</sup>*(*k*)2) = ∑

*<sup>a</sup>*∈*<sup>Y</sup>*

exp(*iπ*

exp(−*im<sup>π</sup>* <sup>⋆</sup>*<sup>ε</sup>* <sup>∑</sup>

*<sup>b</sup>*(*k*)2) = ∑

*<sup>a</sup>*∈*<sup>Y</sup>*

1 *<sup>m</sup>* ∑ *<sup>k</sup>*∈*<sup>L</sup>*

*<sup>k</sup>*∈*<sup>L</sup>*

exp(*iπ*

1 *m* <sup>⋆</sup>*<sup>ε</sup>* ∑ *<sup>k</sup>*∈*<sup>L</sup>*

<sup>2</sup> <sup>∑</sup>*a*∈*<sup>Y</sup> gim*(*a*). When each generator *<sup>ε</sup>*′

1 *m* <sup>⋆</sup>*<sup>ε</sup>* ∑ *<sup>k</sup>*∈*<sup>L</sup>*

<sup>⋆</sup>*HH*′<sup>2</sup>

exp(*iπ*

*s*′

*sk* , it reduces to the following:

*b*(*k*)2)

*a*(*k*)2) (104)

*b*(*k*)2)

*a*(*k*)2) (105)

exp(−*im<sup>π</sup>* <sup>⋆</sup>*<sup>ε</sup>* <sup>∑</sup>

*b*(*k*)2)

exp(−*im<sup>π</sup>* <sup>⋆</sup>*<sup>ε</sup>* <sup>∑</sup>

*<sup>k</sup>*∈*<sup>L</sup>*

*a*(*k*)2) (107)

*<sup>k</sup>*∈*<sup>L</sup>*

*a*(*k*)2) (108)

*a*(*k*)2) (106)

*<sup>k</sup>* of *Y*⊥*<sup>ε</sup> k*

$$H^{\frac{H^2}{2}}\left(\sqrt{-m}\exp(i\frac{\pi}{4})\right)^{("H)^2} \sum\_{b\in Y^{\perp\_\varepsilon}} \exp(i\pi \frac{1}{m} \sum\_{k\in L} b(k)^2) = \sum\_{a\in Y} \exp(-im\pi^\*\varepsilon \sum\_{k\in L} a(k)^2) \tag{109}$$

for a negative *m*.

## **4. Quantum field theory and Zeta function**

In this section the quantum field theory is developed by using the double infinitesimal Fourier transform. The propagator for a system of the harmonic oscillators is considered in the quantum field theory.

#### **4.1. Path integral in the quantum field theory**

**Definition4.1.1.** A path integral of *<sup>f</sup>*(<sup>∈</sup> *<sup>A</sup>*) is defined as follows:

$$\sum\_{a \in X} \varepsilon\_0 f(a) \tag{110}$$

with *ε*<sup>0</sup> := (*H*′ )−( <sup>⋆</sup>*H*)<sup>2</sup> <sup>∈</sup> <sup>⋆</sup>(∗**R**).

It is briefly explained that the complexification of the propagator for the harmonic oscillator is represented as the following path integral. In Feynman's formulation of quantum mechanics([2]), the propagator of the one-dimensional harmonic oscillator is the following path integral: *K*(*q*, *q*0, *t*)

$$=\lim\_{n\to\infty} \int\_{\mathbb{R}^n} (\frac{m}{2\pi i \hbar \varepsilon})^{(n+1)/2} \exp\left(\frac{i\varepsilon}{\hbar} \sum\_{j=1}^{n+1} \left(\frac{m}{2} (\frac{\mathbf{x}\_j - \mathbf{x}\_{j-1}}{\varepsilon})^2 - \frac{m}{2} \lambda^2 \mathbf{x}\_j^2\right)\right) d\mathbf{x}\_1 d\mathbf{x}\_2 \cdots d\mathbf{x}\_n \tag{111}$$

where *x*<sup>0</sup> = *q*0, *xn*+<sup>1</sup> = *q*, *ǫ* = *<sup>t</sup> <sup>n</sup>* . In nonstandard analysis, it is known that, for a sequence an,

$$\lim\_{m \to \infty} a\_{\hbar} = a \quad \text{iff} \quad \,^\* a\_w \approx a \tag{112}$$

for any infinite natural number *<sup>ω</sup>* <sup>∈</sup><sup>∗</sup> **<sup>N</sup>** <sup>−</sup> **<sup>N</sup>**, where <sup>∗</sup>*an* is the <sup>∗</sup> extension of {*an*}*n*∈*N*, and <sup>≈</sup> means that <sup>∗</sup>*aw* <sup>−</sup> *<sup>a</sup>* is infinitesimal, that is, the standard part of *aw* is *<sup>a</sup>*, usually denoted by *st*(∗*aw*) = *a* . The standard part of the nonstandard path integral is written as

$$\text{sst} \int\_{\cdot \mathbb{R}^{w}} (\frac{m}{2\pi i \hbar \varepsilon})^{(\omega+1)/2} \exp\left(\frac{i\varepsilon}{\hbar} \sum\_{j=1}^{w+1} \left(\frac{m}{2} (\frac{\mathbf{x}\_{j} - \mathbf{x}\_{j-1}}{\varepsilon})^{2} - \frac{m}{2} \lambda^{2} \mathbf{x}\_{j}^{2}\right)\right) d\mathbf{x}\_{1} d\mathbf{x}\_{2} \cdots d\mathbf{x}\_{w}.\tag{113}$$

By extending *t* to a complex number, the path integral is complexified to the following :

$$\text{sst} \int\_{\cdot \mathbf{R}^{w}} (\frac{m}{2\pi i \hbar \varepsilon})^{(\omega+1)/2} \exp\left(\frac{i\varepsilon}{\hbar} \sum\_{j=1}^{w+1} \left(\frac{m}{2} (\frac{\mathbf{x}\_{j} - \mathbf{x}\_{j-1}}{\varepsilon})^{2} - \frac{m}{2} \lambda^{2} \mathbf{x}\_{j}^{2}\right)\right) d\mathbf{x}\_{1} d\mathbf{x}\_{2} \cdots d\mathbf{x}\_{w}.\tag{114}$$

**Theorem 4.1.2.** Let *<sup>t</sup>* <sup>∈</sup> **<sup>R</sup>**, *<sup>t</sup>* �<sup>=</sup> *st*(<sup>±</sup> √2*n λ* <sup>1</sup> <sup>−</sup> cos( *<sup>k</sup><sup>π</sup> <sup>ω</sup>*+<sup>1</sup> )), *<sup>k</sup>* <sup>=</sup> 1, 2, ··· , *<sup>ω</sup>*, or for *<sup>t</sup>* <sup>∈</sup> **<sup>C</sup>**, whose imaginary part is negative. The complexified one-dimensional harmonic oscillator standard functional integral is given by ( *<sup>m</sup>* <sup>2</sup>*πih*¯ ) 1 2 *λ* sin(*λt*) exp(*im <sup>h</sup>*¯ *<sup>λ</sup>* sin *<sup>λ</sup>t*((*q*<sup>2</sup> <sup>0</sup> <sup>+</sup> *<sup>q</sup>*2) cos *<sup>λ</sup><sup>t</sup>* <sup>−</sup> <sup>2</sup>*qq*0)).

**Proof.** If *<sup>t</sup>* �<sup>=</sup> *st*(<sup>±</sup> √2*n λ* <sup>1</sup> <sup>−</sup> cos( *<sup>k</sup><sup>π</sup> <sup>ω</sup>*+<sup>1</sup> )), *<sup>k</sup>* <sup>=</sup> 1, 2, ··· , *<sup>ω</sup>*, then

$$t \neq \pm \frac{\sqrt{2}n}{\lambda} \sqrt{1 - \cos(\frac{k\pi}{\omega + 1})}, k = 1, 2, \cdots, \omega \tag{115}$$

for arbitrary infinite number *ω* . The theorem is followed from the discrete calculation using the matrix representation of the operator(cf. [7]).

It corresponds to the well-known real propagator for one dimensional harmonic oscillator. For the d-dimensional harmonic oscillator, *d*-dimensional vectors are written as **q0**, **q** , the square norms are <sup>|</sup>**q0**<sup>|</sup> 2, <sup>|</sup>**q**<sup>|</sup> 2, and the inner product of **q0**, **q** is **q0q**. We have :

**Corollary 4.1.3.** For the complexified d-dimensional harmonic oscillator standard functional integral, the complexified propagator is given by

$$\langle \frac{m}{2\pi i\hbar} \rangle^{\frac{\ell}{2}} (\frac{\lambda}{\sin(\lambda t)})^{\frac{\ell}{2}} \exp(\frac{im}{\hbar} \frac{\lambda}{\sin\lambda t} ((|\mathbf{q}\_{\mathbf{0}}|^{2} + |\mathbf{q}|^{2}) \cos\lambda t - 2\mathbf{q}\mathbf{q}\_{\mathbf{0}})).\tag{116}$$

**Proof.** By factorizing Theorem 4.1.2 into a product on d dimensional, the corollary is obtained.

The trace of the compiexified propagator is calculated for one dimensional harmonic oscillator.

Since

$$\int\_{-\infty}^{\infty} (\frac{m}{2\pi i \hbar})^{\frac{1}{2}} \sqrt{\frac{\lambda}{\sin(\lambda t)}} \exp(\frac{im}{\hbar} \frac{\lambda}{\sin \lambda t} 2((\cos \lambda t - 1)q^2)) dq = \frac{1}{2i \sin(\lambda t/2)}\tag{117}$$

the following is obtained (cf.[6],[7]):

**Theorem 4.1.4.** Let *<sup>t</sup>* <sup>∈</sup> **<sup>R</sup>**, *<sup>t</sup>* �<sup>=</sup> <sup>±</sup>*st*( √2*ω λ* <sup>1</sup> <sup>−</sup> cos( *<sup>k</sup><sup>π</sup> <sup>ω</sup>*+<sup>1</sup> )), *<sup>k</sup>* <sup>=</sup> 1, 2, ··· , *<sup>ω</sup>*, or *<sup>t</sup>* <sup>∈</sup> **<sup>C</sup>**, whose imaginary part is negative. The trace of the complexified one-dimensional harmonic oscillator standard functional integral is given by <sup>1</sup> <sup>2</sup>*<sup>i</sup>* sin(*λt*/2).

**Proof.** By putting *q*<sup>0</sup> = *q* in ( *<sup>m</sup>* <sup>2</sup>*πih*¯ ) 1 2 *λ* sin(*λt*) exp(*im <sup>h</sup>*¯ *<sup>λ</sup>* sin *<sup>λ</sup>t*((*q*<sup>2</sup> <sup>0</sup> <sup>+</sup> *<sup>q</sup>*2) cos *<sup>λ</sup><sup>t</sup>* <sup>−</sup> <sup>2</sup>*qq*0)), the trace is the following integral :

$$\int\_{-\infty}^{\infty} (\frac{m}{2\pi i \hbar})^{\frac{1}{2}} \sqrt{\frac{\lambda}{\sin(\lambda t)}} \exp(\frac{im}{\hbar} \frac{\lambda}{\sin \lambda t} 2((\cos \lambda t - 1)q^2)) dq = \frac{1}{2i \sin(\lambda t/2)}.\tag{118}$$

**Corollary 4.1.5.** If the potential is modified to *V*(*q*) = *<sup>m</sup>* <sup>2</sup> ( *<sup>λ</sup>*<sup>2</sup> <sup>2</sup> <sup>|</sup>*q*<sup>|</sup> 2 − *λ* <sup>2</sup> ), then the trace is 1 <sup>2</sup>*<sup>i</sup>* sin(*λt*/2) exp( *<sup>λ</sup><sup>t</sup>* <sup>2</sup> )).

For the d-dimensional harmonic oscillator, the following is obtained :

**Corollary 4.1.6.** For the trace of the modified complexified propagator for d-dimensional harmonic oscillator, the trace is ( <sup>1</sup> <sup>2</sup>*<sup>i</sup>* sin(*λt*/2) exp( *<sup>λ</sup><sup>t</sup>* <sup>2</sup> )))*d*.

In the next section, Corollaries 4.1.5 and 4.1.6 are used to treat an infinite dimensional harmonic oscillator.

#### **4.2. Representation of the zeta function.**

26 ime knjige

*st* 

**Proof.** If *<sup>t</sup>* �<sup>=</sup> *st*(<sup>±</sup>

square norms are <sup>|</sup>**q0**<sup>|</sup>

obtained.

oscillator.

 ∞ −∞ ( *<sup>m</sup>* 2*πih*¯ ) 1 2 

the following is obtained (cf.[6],[7]):

**Theorem 4.1.4.** Let *<sup>t</sup>* <sup>∈</sup> **<sup>R</sup>**, *<sup>t</sup>* �<sup>=</sup> <sup>±</sup>*st*(

Since

( *<sup>m</sup>* 2*πih*¯ ) *d* <sup>2</sup> ( *<sup>λ</sup>* sin(*λt*) ) *d* <sup>2</sup> exp(

∗**R***<sup>w</sup>* ( *<sup>m</sup>* 2*πih*¯ *ǫ*

**Theorem 4.1.2.** Let *<sup>t</sup>* <sup>∈</sup> **<sup>R</sup>**, *<sup>t</sup>* �<sup>=</sup> *st*(<sup>±</sup>

functional integral is given by ( *<sup>m</sup>*

√2*n λ* 

)(*ω*+1)/2 exp

 *iǫ h*¯

<sup>2</sup>*πih*¯ ) 1 2 *λ*

<sup>1</sup> <sup>−</sup> cos( *<sup>k</sup><sup>π</sup>*

√2*n λ*

*<sup>t</sup>* �<sup>=</sup> <sup>±</sup>

the matrix representation of the operator(cf. [7]).

2, <sup>|</sup>**q**<sup>|</sup>

integral, the complexified propagator is given by

*λ* sin(*λt*) exp(

oscillator standard functional integral is given by <sup>1</sup>

*w*+1 ∑ *j*=1

√2*n λ* 

 *m* 2 (

By extending *t* to a complex number, the path integral is complexified to the following :

*xj* <sup>−</sup> *xj*−<sup>1</sup>

<sup>1</sup> <sup>−</sup> cos( *<sup>k</sup><sup>π</sup>*

sin(*λt*) exp(*im*

imaginary part is negative. The complexified one-dimensional harmonic oscillator standard

*<sup>ω</sup>*+<sup>1</sup> )), *<sup>k</sup>* <sup>=</sup> 1, 2, ··· , *<sup>ω</sup>*, then

*ω* + 1

for arbitrary infinite number *ω* . The theorem is followed from the discrete calculation using

It corresponds to the well-known real propagator for one dimensional harmonic oscillator. For the d-dimensional harmonic oscillator, *d*-dimensional vectors are written as **q0**, **q** , the

**Corollary 4.1.3.** For the complexified d-dimensional harmonic oscillator standard functional

**Proof.** By factorizing Theorem 4.1.2 into a product on d dimensional, the corollary is

The trace of the compiexified propagator is calculated for one dimensional harmonic

*λ* sin *λt*

*im h*¯

*im h*¯

*λ* sin *λt*

> √2*ω λ*

whose imaginary part is negative. The trace of the complexified one-dimensional harmonic

2, and the inner product of **q0**, **q** is **q0q**. We have :

((|**q0**<sup>|</sup>

<sup>2</sup> <sup>+</sup> <sup>|</sup>**q**<sup>|</sup>

<sup>2</sup>((cos *<sup>λ</sup><sup>t</sup>* <sup>−</sup> <sup>1</sup>)*q*2))*dq* <sup>=</sup> <sup>1</sup>

<sup>1</sup> <sup>−</sup> cos( *<sup>k</sup><sup>π</sup>*

<sup>2</sup>*<sup>i</sup>* sin(*λt*/2).

<sup>1</sup> <sup>−</sup> cos( *<sup>k</sup><sup>π</sup>*

*<sup>ǫ</sup>* )<sup>2</sup> <sup>−</sup> *<sup>m</sup>*

*<sup>h</sup>*¯ *<sup>λ</sup>* sin *<sup>λ</sup>t*((*q*<sup>2</sup>

<sup>2</sup> *<sup>λ</sup>*2*x*<sup>2</sup> *j* 

*<sup>ω</sup>*+<sup>1</sup> )), *<sup>k</sup>* <sup>=</sup> 1, 2, ··· , *<sup>ω</sup>*, or for *<sup>t</sup>* <sup>∈</sup> **<sup>C</sup>**, whose

<sup>0</sup> <sup>+</sup> *<sup>q</sup>*2) cos *<sup>λ</sup><sup>t</sup>* <sup>−</sup> <sup>2</sup>*qq*0)).

), *<sup>k</sup>* <sup>=</sup> 1, 2, ··· , *<sup>ω</sup>* (115)

<sup>2</sup>) cos *<sup>λ</sup><sup>t</sup>* <sup>−</sup> <sup>2</sup>**qq**0)). (116)

<sup>2</sup>*<sup>i</sup>* sin(*λt*/2) (117)

*<sup>ω</sup>*+<sup>1</sup> )), *<sup>k</sup>* <sup>=</sup> 1, 2, ··· , *<sup>ω</sup>*, or *<sup>t</sup>* <sup>∈</sup> **<sup>C</sup>**,

*dx*1*dx*<sup>2</sup> ··· *dxw*. (114)

Corollary 4.1.5 is extended to an infinite dimensional harmonic oscillator using nonstandard analysis. For it the three types of extension ∗**R** , <sup>⋆</sup>∗**R**, #⋆∗**<sup>R</sup>** of **<sup>R</sup>** are prepared corresponding to Definition 2.2.2 , then the three stages of infinite numbers exist. In these three extension fields, we fix infinite natural numbers *HF* <sup>∈</sup> <sup>∗</sup>**<sup>N</sup>** ,*HT* <sup>∈</sup> <sup>⋆</sup>∗**N**, *<sup>H</sup>*" <sup>∈</sup> 2#<sup>⋆</sup>∗**<sup>N</sup>** . Let *<sup>T</sup>* be a positive standard real number and let *<sup>ǫ</sup><sup>T</sup>* , *<sup>ǫ</sup>*" be infinitesimals in <sup>⋆</sup>∗**<sup>R</sup>** , #⋆∗**<sup>R</sup>** defined by *<sup>T</sup> HT* , <sup>1</sup> *<sup>H</sup>*" . A lattice *L*" and two function space *X*, *A* are defined as the following:

$$L'' := \left\{ \varepsilon'' z'' \, \Big| \, z'' \in \, ^{\#\ast} \mathbf{Z}\_{\prime} - \frac{\mathbf{H}''}{2} \le \prime'' \mathbf{z}'' < \frac{\mathbf{H}''}{2} \right\}.$$

$$X := \left\{ a : ^{\#\ast} \left\{ 0, 1, \cdot \cdot \cdot . H\_{\mathrm{F}} - 1 \right\} \to L'', \mathrm{internal} \right\}.$$

$$A := \left\{ a : ^{\#} \left\{ 0, 1, \cdot \cdot \cdot . H\_{\mathrm{T}} \right\} \to \mathrm{X}, \mathrm{internal} \right\}.$$

Then an element *a* of *A* is written as the component (*a<sup>k</sup> <sup>j</sup>* , 0 <sup>≤</sup> *<sup>j</sup>* <sup>≤</sup> *HT*, 0 <sup>≤</sup> *<sup>k</sup>* <sup>≤</sup> *HF* <sup>−</sup> <sup>1</sup>). All prime numbers are ordered as *p*(1) = 2, *p*(2) = 3, ... , *p*(*n*) < *p*(*n* + 1), ... , that is, *p* is a mapping from **<sup>N</sup>** to the set of prime numbers, *<sup>p</sup>* : **<sup>N</sup>** → {prime number} . Let *<sup>λ</sup><sup>k</sup>* be ln<sup>∗</sup> *<sup>p</sup>*(*k*) for each *<sup>k</sup>* , 0 <sup>≤</sup> *<sup>k</sup>* <sup>≤</sup> *HF* <sup>−</sup> 1. A potential *Vk* : #⋆<sup>∗</sup> **<sup>R</sup>** <sup>→</sup>#⋆<sup>∗</sup> **<sup>R</sup>** is defined for each *<sup>k</sup>* , <sup>0</sup> <sup>≤</sup> *<sup>k</sup>* <sup>≤</sup> *HF* <sup>−</sup> 1, as *Vk*(*q*) = *<sup>λ</sup>*<sup>2</sup> *k* <sup>2</sup> <sup>|</sup>*q*<sup>|</sup> 2 − *λk* <sup>2</sup> . An element *<sup>α</sup>* of *<sup>X</sup>* is written as the component *<sup>α</sup>* = (*αk*, 0 <sup>≤</sup> *<sup>k</sup>* <sup>≤</sup> *HF* <sup>−</sup> <sup>1</sup>).

Let *V* be a global potential as the following:

$$V(\boldsymbol{a}) = \sum\_{k=0}^{H\_F - 1} V\_k(\boldsymbol{a}^k) \left( = \sum\_{k=0}^{H\_F - 1} \left( \frac{\lambda\_k^2}{2} \left| \boldsymbol{a}^k \right|^2 - \frac{\lambda\_k}{2} \right) \right). \tag{119}$$

In order to transport *<sup>t</sup>* to later, the element <sup>−</sup>*λ<sup>k</sup>* <sup>2</sup> is put in the usual potential for harmonic oscillators. It is considered the following summation *<sup>K</sup>*(*a*, *<sup>b</sup>*, *<sup>t</sup>*) depending of *<sup>a</sup>*, *<sup>b</sup>* <sup>∈</sup> *<sup>X</sup>*:

$$K(a,b,t) = \sum\_{a \in A, a\_0 = a.a\_{H\_T} = b} ((\varepsilon')^{H\_F})^{H\_T} ((\frac{1}{2\pi \varepsilon\_T})^{H\_F})^{H\_T} \exp(\varepsilon\_T (\frac{1}{2} \sum\_{j=1}^{H\_T} |\frac{a\_j - a\_{j-1}}{\varepsilon\_T}|^2 - V(a\_{\parallel}))). \tag{120}$$

Then *K*(*a*, *b*, *t*) is calculated,

$$\mathcal{K}(a,b,t) = (\frac{1}{2\pi\epsilon\_T})^{H\_T} \prod\_{k=1}^{H\_T-1} \sum\_{\substack{d\_j^k \in \mathcal{U}, 0 \le j \le H\_T-1}} (\boldsymbol{\epsilon}')^{H\_T} (\frac{1}{2\pi\epsilon\_T})^{H\_T} \exp(\boldsymbol{\epsilon}\_T (\frac{1}{2} \sum\_{j=1}^{H\_T} |\frac{a\_j^k - a\_{j-1}^k}{\boldsymbol{\epsilon}\_T}|^2 - V(\boldsymbol{d}\_j^k))), \tag{121}$$

where *a*<sup>0</sup> = *a*, *aHT* = *b*.

The summation <sup>∑</sup>*a*∈*X*(*ǫ*′ )*HTK*(*a*, *a*.*t*) is denoted by *tr*(*K*(*a*, *a*, *t*)). Three correspondences putting standard parts are written as *st*# : #⋆∗**<sup>R</sup>** <sup>→</sup> <sup>⋆</sup>∗**R**, *st*<sup>⋆</sup> : <sup>⋆</sup>∗**<sup>R</sup>** <sup>→</sup> <sup>∗</sup>**R**, *st*<sup>∗</sup> : <sup>∗</sup>**<sup>R</sup>** <sup>→</sup> **<sup>R</sup>** . When there are no confusion, they are simply written as *st*. The composition *st*<sup>∗</sup> ◦ *st*<sup>⋆</sup> ◦ *st*# : #⋆∗**<sup>R</sup>** <sup>→</sup> **<sup>R</sup>** is denoted also by *st* for simplicity.

**Theorem 4.2.1.** If the real part of *t* is greater than 1, the standard part *st*(*tr*(*K*(*a*, *a*, *t*))) of *tr*(*K*(*a*, *a*, *t*)) corresponds to Riemann's zeta function *ζ*(*t*).

Proot. The standard parts of *tr*(*K*(*a*, *a*, *t*)) as follows.

$$\begin{aligned} st\_{\#}(\operatorname{tr}(\operatorname{K}(a,a,t))) &= \\ \prod\_{k=1}^{H\_{\mathbb{F}}-1} \int \int \cdots \int (\frac{1}{2\pi \mathfrak{e}\_{\mathbb{T}}})^{H\_{\mathbb{T}}} \exp(\epsilon \tau\_{\mathbb{T}}(\frac{1}{2} \sum\_{j=1}^{H\_{\mathbb{T}}} (\frac{q\_{j}^{k}-q\_{j-1}^{k}}{\mathfrak{e}\_{\mathbb{T}}})^{2} - V(q\_{j}^{k}))) dq\_{0}^{k} dq\_{1}^{k} \cdots dq\_{H\mathbb{\tau}-1}^{k} \end{aligned} \tag{122}$$

$$=\prod\_{k=1}^{H\_{\rm F}-1} \int \{ \int \cdots \int (\frac{1}{2\pi\epsilon\_{\rm T}})^{H\_{\rm T}} \exp(\epsilon\_{\rm T}(\frac{1}{2}\sum\_{j=1}^{H\_{\rm T}}(\frac{q\_j^k - q\_{j-1}^k}{\epsilon\_{\rm T}})^2 - V(q\_j^k))) dq\_1^k \cdots dq\_{H\_{\rm T}-1}^k \} dq\_0^k \tag{123}$$

by Fubini's theorem. Furthermore,

$$= \prod\_{k=1}^{H\_{\rm T}-1} \text{st}\_{\star} \{ \int \dots \int (\frac{1}{2\pi \epsilon\_{\rm T}})^{H\_{\rm T}} \exp(\epsilon\_{\rm T} \{ \frac{1}{2} \sum\_{j=1}^{H\_{\rm T}} (\frac{q\_j^k - q\_{j-1}^k}{\epsilon\_{\rm T}})^2 - V(q\_j^k)) dq\_1^k \cdots dq\_{H\_{\rm T}-1}^k \} dq\_0^k \tag{124}$$

by the same calculation of Theorem 4.1.2 (cf.[6],[7]) ,

$$=\prod\_{k=0}^{H\_F-1} (\frac{1}{2i\sin(\frac{\lambda\_k t}{2i})} \exp(\frac{\lambda\_k t}{2})) = \prod\_{k=0}^{H\_F-1} \frac{1}{1 - p\_k^{-t}}.\tag{125}$$

By Lebesque's convergence theorem, (*st*∗(*st*⋆(*st*#(*tr*(*K*(*a*, *<sup>a</sup>*, *<sup>t</sup>*))))) = <sup>∏</sup><sup>∞</sup> *k*=0 1 <sup>1</sup>−*p*−*<sup>t</sup> k* = *ζ*(*t*) , if the real part of *t* is positive .

#### **4.2. Another representation of the zeta function**

In this section, both *st*∗,*st*<sup>⋆</sup> and *st*# are denoted as *st* for the simplification. A functional is defined on *X*, and a relationship between the functional and Riemann's zeta function is shown later. The nonstandard extension <sup>∗</sup> *<sup>p</sup>* : <sup>∗</sup>**<sup>N</sup>** <sup>→</sup> ∗{prime number} is written as <sup>∗</sup> *<sup>p</sup>*([*lµ*]) = [*p*(*lµ*)], and a mapping *<sup>p</sup>*˜ : <sup>∗</sup>**<sup>N</sup>** <sup>→</sup> <sup>⋆</sup>( ∗{prime number}) is defined as *<sup>p</sup>*˜([*lµ*]) = <sup>⋆</sup>[*p*(*lµ*)]. For *<sup>s</sup>* <sup>∈</sup> **<sup>C</sup>**, *Zs*(<sup>∈</sup> *<sup>A</sup>*) is defined as the following :

$$Z\_s(a) := \prod\_{k \in L} \vec{p}(H(k + \frac{H}{2}) + 1)^{(-s(a(k) + \frac{H'}{2}))}.\tag{126}$$

Now *H*(*k* + *<sup>H</sup>* <sup>2</sup> ) + 1 is an element of <sup>∗</sup>**N** and *a*(*k*) + *H*′ /2 is an element of <sup>⋆</sup>( <sup>∗</sup>**N**). Then *Zs*(*a*) is calculated as exp(−*<sup>s</sup>* <sup>∑</sup>*k*∈*<sup>L</sup>* log(*p*˜(*H*(*<sup>k</sup>* <sup>+</sup> *<sup>H</sup>* <sup>2</sup> ) + <sup>1</sup>))*a*(*k*)) <sup>∏</sup>*k*∈*<sup>L</sup> <sup>p</sup>*˜(*H*(*<sup>k</sup>* <sup>+</sup> *<sup>H</sup>* <sup>2</sup> ) + <sup>1</sup>)−*<sup>s</sup> <sup>H</sup>*′ 2 . The following theorem is obtained for the Fourier transform of *Zs* for Definition 2.2 1: **Theorem 4.3.1.**

$$(F((Z\_s))(b) = \left(\prod\_{k \in L} \mathfrak{p}(H(k + \frac{H}{2}) + 1)\right)^{-s\frac{H'}{2}}$$

$$+\prod\_{k\in L} \epsilon' \frac{\sinh((2\pi i b(k) + s\log \tilde{p}(H(k + \frac{H}{2}) + 1))\frac{H'}{2})}{\exp(-\frac{\zeta'}{2}(2\pi i b(k) + s\log \tilde{p}(H(k + \frac{H}{2}) + 1))\sinh(\frac{\zeta'}{2}(2\pi i b(k) + s\log \tilde{p}(H(k + \frac{H}{2}) + 1))}.\tag{127}$$

**Proof.**

28 ime knjige

*K*(*a*, *b*, *t*) = ∑

*<sup>K</sup>*(*a*, *<sup>b</sup>*, *<sup>t</sup>*)=( <sup>1</sup>

where *a*<sup>0</sup> = *a*, *aHT* = *b*.

The summation <sup>∑</sup>*a*∈*X*(*ǫ*′

*HF*−<sup>1</sup> ∏ *k*=1

= *HF*−<sup>1</sup> ∏ *k*=1

> = *HF*−<sup>1</sup> ∏ *k*=1 *st*⋆{ { ··· ( <sup>1</sup> 2*πǫ<sup>T</sup>*

 { ··· ( <sup>1</sup> 2*πǫ<sup>T</sup>*

··· ( <sup>1</sup> 2*πǫ<sup>T</sup>*

by Fubini's theorem. Furthermore,

Then *K*(*a*, *b*, *t*) is calculated,

2*πǫ<sup>T</sup>* )*HT HF*−<sup>1</sup> ∏ *k*=1

In order to transport *<sup>t</sup>* to later, the element <sup>−</sup>*λ<sup>k</sup>*

((*ǫ*′

*ak*

*tr*(*K*(*a*, *a*, *t*)) corresponds to Riemann's zeta function *ζ*(*t*).

)*HT* exp(*ǫT*(

)*HT* exp(*ǫT*(

)*HT* exp(*ǫT*(

by the same calculation of Theorem 4.1.2 (cf.[6],[7]) ,

1 2

> 1 2

*HT* ∑ *j*=1 ( *qk <sup>j</sup>* <sup>−</sup> *<sup>q</sup><sup>k</sup> j*−1 *ǫT*

1 2 *HT* ∑ *j*=1 ( *qk <sup>j</sup>* <sup>−</sup> *<sup>q</sup><sup>k</sup> j*−1 *ǫT*

*HT* ∑ *j*=1 ( *qk <sup>j</sup>* <sup>−</sup> *<sup>q</sup><sup>k</sup> j*−1 *ǫT*

Proot. The standard parts of *tr*(*K*(*a*, *a*, *t*)) as follows.

*<sup>a</sup>*∈*A*,*a*0=*a*.*aHT* <sup>=</sup>*<sup>b</sup>*

oscillators. It is considered the following summation *<sup>K</sup>*(*a*, *<sup>b</sup>*, *<sup>t</sup>*) depending of *<sup>a</sup>*, *<sup>b</sup>* <sup>∈</sup> *<sup>X</sup>*:

2*πǫ<sup>T</sup>*

(*ǫ*′

)*HT* ( <sup>1</sup> 2*πǫ<sup>T</sup>*

putting standard parts are written as *st*# : #⋆∗**<sup>R</sup>** <sup>→</sup> <sup>⋆</sup>∗**R**, *st*<sup>⋆</sup> : <sup>⋆</sup>∗**<sup>R</sup>** <sup>→</sup> <sup>∗</sup>**R**, *st*<sup>∗</sup> : <sup>∗</sup>**<sup>R</sup>** <sup>→</sup> **<sup>R</sup>** . When there are no confusion, they are simply written as *st*. The composition *st*<sup>∗</sup> ◦ *st*<sup>⋆</sup> ◦ *st*# : #⋆∗**<sup>R</sup>** <sup>→</sup> **<sup>R</sup>** is denoted also by *st* for simplicity. **Theorem 4.2.1.** If the real part of *t* is greater than 1, the standard part *st*(*tr*(*K*(*a*, *a*, *t*))) of

)*HF* )*HT* exp(*ǫT*(

)*HT* exp(*ǫT*(

)*HTK*(*a*, *a*.*t*) is denoted by *tr*(*K*(*a*, *a*, *t*)). Three correspondences

)<sup>2</sup> <sup>−</sup> *<sup>V</sup>*(*q<sup>k</sup>*

)<sup>2</sup> <sup>−</sup> *<sup>V</sup>*(*q<sup>k</sup>*

)<sup>2</sup> <sup>−</sup> *<sup>V</sup>*(*q<sup>k</sup>*

*j*)))*dq<sup>k</sup>* 0*dq<sup>k</sup>*

*j*)))*dq<sup>k</sup>*

*j*)))*dq<sup>k</sup>*

1 2

*HT* ∑ *j*=1 |

1 2 *HT* ∑ *j*=1 | *ak <sup>j</sup>* <sup>−</sup> *<sup>a</sup><sup>k</sup> j*−1 *ǫT* | 2 <sup>−</sup> *<sup>V</sup>*(*a<sup>k</sup>*

)*HF* )*HT* (( <sup>1</sup>

∑

*<sup>j</sup>* <sup>∈</sup>*L*′,0≤*j*≤*HT*−<sup>1</sup>

<sup>2</sup> is put in the usual potential for harmonic

*aj* <sup>−</sup> *aj*−<sup>1</sup> *ǫT*


*st*#(*tr*(*K*(*a*, *a*, *t*))) =

<sup>1</sup> ··· *dq<sup>k</sup>*

*st*⋆*st*#(*tr*(*K*(*a*, *a*, *t*))) =

*HT*−1}*dq<sup>k</sup>*

<sup>1</sup> ··· *dq<sup>k</sup>*

<sup>1</sup> ··· *dq<sup>k</sup>*

*HT*−<sup>1</sup> (122)

<sup>0</sup> (123)

<sup>0</sup>} (124)

*HT*−1}*dq<sup>k</sup>*

<sup>−</sup> *<sup>V</sup>*(*aj*))).(120)

*<sup>j</sup>*))), (121)

$$\begin{aligned} (\mathsf{F}((\mathsf{Z}\_{s}))(b) &= \left(\prod\_{k \in L} \not{\mathfrak{p}}(H(k + \frac{H}{2}) + 1)\right)^{-s\frac{H}{2}} \\ \cdot \sum\_{a \in X} \varepsilon\_{0} \exp\left(-s \sum\_{k \in L} \log \not{\mathfrak{p}}(H(k + \frac{H}{2}) + 1)a(k)\right) \exp\left(-2\pi i \sum\_{k \in L} a(k)b(k)\right) \\ = \left(\prod\_{k \in L} \not{\mathfrak{p}}(H(k + \frac{H}{2}) + 1)\right)^{-s\frac{H^{\sharp}}{2}} \cdot \sum\_{a \in X} \varepsilon\_{0} \exp\left(-(2\pi i \not{b}(k) + s \log \not{\mathfrak{p}}(H(k + \frac{H}{2}) + 1))a(k)\right) \\ = \left(\prod\_{k \in L} \not{\mathfrak{p}}(H(k + \frac{H}{2}) + 1)\right)^{-s\frac{H^{\sharp}}{2}} \end{aligned}$$

$$+\prod\_{k\in L} \varepsilon' \frac{\sinh((2\pi i \, b(k) + s \log \, \mathfrak{p}(H(k + \frac{H}{2}) + 1))\frac{H}{2})}{\exp\left(-\frac{\varepsilon'}{2}(2\pi i \, b(k) + s \log \mathfrak{p}(H(k + \frac{H}{2}) + 1))\sinh\left(\frac{\varepsilon'}{2}(2\pi i \, b(k) + s \log \mathfrak{p}(H(k + \frac{H}{2}) + 1))\right)\right)}.$$

Riemann's zeta function *ζ*(*s*) is defined by *ζ*(*s*) = ∏<sup>∞</sup> *l*=1 1 <sup>1</sup>−*p*(*l*)−*<sup>s</sup>* for Re(*s*) <sup>&</sup>gt; 1. Let *<sup>Y</sup>***<sup>Z</sup>** be a subgroup of *X* so that each generator of (*Y***Z**)*<sup>k</sup>* is equal to 1. Then the following theorem is obtained :

**Theorem 4.3.2.** If Re(*s*) <sup>&</sup>gt; 1, then st(st(∑*a*∈*Y***<sup>Z</sup>** (*Zs*))(*a*))) = *<sup>ζ</sup>*(*s*).

**Proof.** st(st(∑*a*∈*Y***<sup>Z</sup>** (*Zs*)(*a*))) =st st<sup>∏</sup>*k*∈*<sup>L</sup> <sup>p</sup>*˜(*H*(*<sup>k</sup>* <sup>+</sup> *<sup>H</sup>* <sup>2</sup> ) + 1) (−*s*(*a*(*k*)+ *<sup>H</sup>*′ <sup>2</sup> ))

$$\tilde{\rho} = st \left( st \left( \prod\_{k \in L} \frac{1 - \tilde{p}(H(k + \frac{H}{2}) + 1)^{-sH'}}{1 - \tilde{p}(H(k + \frac{H}{2}) + 1)^{-s}} \right) \right) = st \left( \prod\_{k \in L} \frac{1}{1 - \tilde{p}(H(k + \frac{H}{2}) + 1)^{-s}} \right) = \zeta(s). \tag{128}$$

Furthermore, Poisson summation formula and Theorem 4.3.2 imply the following : **Corollary 4.3.3.**

$$\text{st}(\sum\_{b \in Y\_{\mathbb{Z}}^{\perp}} (F(Z\_s)(b)) = \text{st}\left(\prod\_{k \in L} \frac{1 - \tilde{\rho}(H(k + \frac{H}{2}) + 1)^{-sH'}}{1 - \tilde{\rho}(H(k + \frac{H}{2}) + 1)^{-s}}\right). \tag{129}$$

Hence we obtain :

$$\text{st}(\text{st}(\sum\_{b \in \mathcal{Y}\_{\mathbf{Z}}^{\perp}} (F(Z\_s)(b)))) = \zeta(s) \tag{130}$$

for Re(*s*) > 1.

In general, the physical theory has variables for position, time, and fields. Especially there are many kinds of variables in quantum field theory. The function depends on such variables mixed as *f*(**q**, *t*, **a**, **b**, **c**) where **q** is position, *t* is time, **a**, **b**, **c** are fields. When the function is treated for such mixed variables, the Kinoshita's infinitesimal Fourier transform and our double infinitesimal Fourier transform are applied in the double extended number field. The two kinds of Fourier transforms can be used for one function. In the theory, the delta functions for variable and for fields have different infinitesimals and infinite values. The delta function for fields has an infinitesimal much smaller and much bigger infinite number. However they can be treat in the double extended number field. Two kinds of delta functions are defined with another degrees. One delta function has an infinitesimal of the first degree and the other delta function has an infinitesimal of the second degree. The infinitesimal of the second one can not be observable with respect to the first one.

10.5772/59963

## **5. Conclusion**

30 ime knjige

· ∏ *<sup>k</sup>*∈*<sup>L</sup> ε*

obtained :

= *st st* ∏ *<sup>k</sup>*∈*<sup>L</sup>*

**Corollary 4.3.3.**

Hence we obtain :

for Re(*s*) > 1.

exp(<sup>−</sup> *<sup>ε</sup>*′

**Proof.** st(st(∑*a*∈*Y***<sup>Z</sup>** (*Zs*)(*a*))) =st

<sup>1</sup> <sup>−</sup> *<sup>p</sup>*˜(*H*(*<sup>k</sup>* <sup>+</sup> *<sup>H</sup>*

*st*( ∑ *<sup>b</sup>*∈*Y*<sup>⊥</sup> **Z**

<sup>1</sup> <sup>−</sup> *<sup>p</sup>*˜(*H*(*<sup>k</sup>* <sup>+</sup> *<sup>H</sup>*

′ sinh((2*πi b*(*k*) + *<sup>s</sup>*log *<sup>p</sup>*˜(*H*(*<sup>k</sup>* <sup>+</sup> *<sup>H</sup>*

**Theorem 4.3.2.** If Re(*s*) <sup>&</sup>gt; 1, then st(st(∑*a*∈*Y***<sup>Z</sup>** (*Zs*))(*a*))) = *<sup>ζ</sup>*(*s*).

 st 

<sup>2</sup> ) + <sup>1</sup>)−*sH*′

<sup>2</sup> ) + <sup>1</sup>)−*<sup>s</sup>*

(*F*(*Zs*)(*b*)) = *st*

*st*(*st*( ∑ *<sup>b</sup>*∈*Y*<sup>⊥</sup> **Z**

the second one can not be observable with respect to the first one.

<sup>2</sup> (2*πi b*(*k*) + *<sup>s</sup>*log *<sup>p</sup>*˜(*H*(*<sup>k</sup>* <sup>+</sup> *<sup>H</sup>*

Riemann's zeta function *ζ*(*s*) is defined by *ζ*(*s*) = ∏<sup>∞</sup>

<sup>2</sup> ) + <sup>1</sup>)) *<sup>H</sup>*′ 2 )

1

*l*=1

<sup>2</sup> ) + 1)

<sup>1</sup> <sup>−</sup> *<sup>p</sup>*˜(*H*(*<sup>k</sup>* <sup>+</sup> *<sup>H</sup>*

<sup>1</sup> <sup>−</sup> *<sup>p</sup>*˜(*H*(*<sup>k</sup>* <sup>+</sup> *<sup>H</sup>*

<sup>2</sup> (2*πi b*(*k*) + *<sup>s</sup>*log *<sup>p</sup>*˜(*H*(*<sup>k</sup>* <sup>+</sup> *<sup>H</sup>*

(−*s*(*a*(*k*)+ *<sup>H</sup>*′

1

<sup>2</sup> ) + <sup>1</sup>)−*sH*′

<sup>2</sup> ) + <sup>1</sup>)−*<sup>s</sup>*

(*F*(*Zs*)(*b*)))) = *ζ*(*s*) (130)

<sup>1</sup> <sup>−</sup> *<sup>p</sup>*˜(*H*(*<sup>k</sup>* <sup>+</sup> *<sup>H</sup>*

<sup>2</sup> ) + <sup>1</sup>)).

<sup>1</sup>−*p*(*l*)−*<sup>s</sup>* for Re(*s*) <sup>&</sup>gt; 1. Let *<sup>Y</sup>***<sup>Z</sup>** be a

<sup>2</sup> ))

= *ζ*(*s*). (128)

. (129)

<sup>2</sup> ) + <sup>1</sup>)−*<sup>s</sup>*

<sup>2</sup> ) + <sup>1</sup>)) sinh(*ε*′

subgroup of *X* so that each generator of (*Y***Z**)*<sup>k</sup>* is equal to 1. Then the following theorem is

<sup>∏</sup>*k*∈*<sup>L</sup> <sup>p</sup>*˜(*H*(*<sup>k</sup>* <sup>+</sup> *<sup>H</sup>*

 = *st* ∏ *<sup>k</sup>*∈*<sup>L</sup>*

Furthermore, Poisson summation formula and Theorem 4.3.2 imply the following :

 ∏ *<sup>k</sup>*∈*<sup>L</sup>*

In general, the physical theory has variables for position, time, and fields. Especially there are many kinds of variables in quantum field theory. The function depends on such variables mixed as *f*(**q**, *t*, **a**, **b**, **c**) where **q** is position, *t* is time, **a**, **b**, **c** are fields. When the function is treated for such mixed variables, the Kinoshita's infinitesimal Fourier transform and our double infinitesimal Fourier transform are applied in the double extended number field. The two kinds of Fourier transforms can be used for one function. In the theory, the delta functions for variable and for fields have different infinitesimals and infinite values. The delta function for fields has an infinitesimal much smaller and much bigger infinite number. However they can be treat in the double extended number field. Two kinds of delta functions are defined with another degrees. One delta function has an infinitesimal of the first degree and the other delta function has an infinitesimal of the second degree. The infinitesimal of The real and complex number fields are extended to the larger number fields where there are many infinitesimal and infinite numbers. A lattice of infinitesimal width is included in the extended real number field. An infinitesimal Fourier transform theory is constructed on the infinitesimal lattice. These extended number fields are furthermore extended to much higher generalized fields where there exist much higher infinitesimal and infinite numbers. A double infinitesimal Fourier transform theory is developed on these double extended number fields. The usual formulae for Fourier theory are satisfied in the theory, especially the Poisson summation formula. The Fourier theory is based on the integral theory for functionals corresponding to the path integral in the physics. The theory is associated to the physical theory in the quantum field theory which is mathematically rigorous . For an application for the double infinitesimal calculation, Riemann's zeta function is represented as such an integral for the propagator of an infinite dimensional harmonic oscillator.

## **Acknowledgement**

Supported by JSPS Grant-in-Aid for Scientific Research (B) No. 25287010.

## **Author details**

Takashi Gyoshin Nitta

Department of Mathematics, Faculty of Education, Faculty of Education, Mie University, Japan

## **References**


**Section 2**

**Physical Sciences**

32 ime knjige

[8] Nitta T, Okada T. Double infinitesimal Fourier transformation for the space of functionals and reformulation of Feynman path integral, Lecture Note Series in

[9] Nitta T, Okada T. Infinitesimal Fourier transformation for the space of functionals.

[10] Nitta T, Okada T. Poisson summation formula for the space of f functionals, Topics in contemporary differential geometry, complex analysis and mathematical physics. ;2007.

[11] Nitta T, Okada T, Tzouvaras A. Classification of non-well-founded sets and an

[12] Nitta T,Péraire Y. Divergent Fourier analysis using degrees of observability. Nonlinear

[14] Péraire Y. Some extensions of the principles of idealization transfer and choice in the

[15] Remmert R. Theory of complex functions. Graduate Texts in Mathematics 122 Springer

[13] Péraire Y. Théorie relative des ensembles internes. Osaka J.Math. 1992; 29 267-297.

[16] Saito M. Ultraproduct and non-standard analysis. Tokyo tosho; 1976 in Japanese.

relative internal set theory, Arch. Math. Logic . 1995; 34 269-277.

[17] Satake I. The temptation to algebra. Yuseisha; 1996 in Japanese.

[18] Takeuti G. Dirac space. Proc. Japan Acad. 1962; 38 414-418.

Mathematics, Osaka University 2002 7 255-298 in Japanese.

application. Math. Log. Quart. ;2003 49 187-200.

Nihonkai Math. J. 2005; 16 1-21.

138 Fourier Transform - Signal Processing and Physical Sciences

Anal. 2009; 71 2462-2468.

Berlin-Heidelberg-New York; 1992.

p261-268.

**Chapter 6**
