**Experimental Data Deconvolution Based on Fourier Transform Applied in Nanomaterial Structure**

Adrian Bot, Nicolae Aldea and Florica Matei

Additional information is available at the end of the chapter

http://dx.doi.org/10.5772/59667

## **1. Introduction**

In many kinds of experimental measurements, such as astrophysics, atomic physics, biophy‐ sics, geophysics, high energy physics, nuclear physics, plasma physics, solid state physics, bending or torsion elastic, heat propagation or statistical mechanics, the signal measured in the laboratory can be expressed mathematically as a convolution of two functions. The first represents the resolution function called the instrumental signal, which is specific for each setup, and the second is the true sample that contains all physical information. These phe‐ nomena can be modelled by an integral equation, which means the unknown function is under the integral operator. The most important type of integral equation applied in physical and technical signal treatments is the Fredholm integral equation of the first kind. The opposite process when used for true sample function determination is known in the literature as experimental data deconvolution. Solution determination of the deconvolution equation does not readily unveil its true mathematical implications concerning the stability of the solutions or other aspects. Thus, from this point of view, the problem is described as improper or illposed. The most rigorous methods for solving the deconvolution equation are: regularization, spline function approximation and Fourier transform technique. The essential feature of regularization method is the replacement of a given improper problem with another, auxiliary, correctly posed problem. The second method consists in approximating both the experimental and instrumental signals by piecewise cubic spline. Most often when using this technique, the true sample function belongs to the same piecewise cubic spline class. The topic of this chapter is the application of Fourier transform in experimental data deconvolution for use in nano‐ material structures.

© 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 reproduction in any medium, provided the original work is properly cited.

## **2. Mathematical background of signal deconvolution**

As mentioned in the introduction, ill-posed problems from a mathematical point of view have many applications in physics and technologies [1]. In addition to the abovementioned examples, the examples below should be noted.

Solving the Cauchy problem for the Laplace equation, *ΔU* =0 has a direct application in biophysics as in [2]. The problem consists in determining the biopotential distribution within the body denoted by *U*, when the body surface potential values are known. The phenomenon

is modelled by the Laplace equation, and the Cauchy conditions are *<sup>U</sup>* <sup>|</sup>*<sup>S</sup>* <sup>=</sup> *<sup>f</sup>* (*S*) and ∂*<sup>U</sup>* <sup>∂</sup>*<sup>n</sup>* <sup>|</sup> *<sup>S</sup>* =0,

where *S* represents the surface of the body.

The determination of radioactive substances in the body, as in [2], and protein crystallography structures also deal with ill-posed problems: see [3].

The same formalism is used in quantum mechanics to determine the particle scattering crosssection on different targets, as well as in plasma physics in the case of the electron distribution after speed is received from the dispersion curve analysis [2].

An intuitive way of grasping an ill-posed problem can be modelled by the movement of the vibrating string when many forces are acting perpendicularly on the string, as represented in Figure 1.

**Figure 1.** Physical model of the vibrating string

In terms of the mathematical equation, the phenomenon described above has a correspondent in physics spectroscopy used in the study of nanomaterials. In the first instance, it is considered that in the point of abscissa *xk*, force *f* → *<sup>k</sup>* acts perpendicularly to the direction of the string. The string movement in the vertical plane at an arbitrary point *s* is given by the proportionality relationship,

$$h(\mathbf{s}) = \mathbf{g}(\mathbf{s}, \mathbf{x}\_k) f(\mathbf{x}\_k) \tag{1}$$

where *g*(*s*, *xk* ) characterizes the impact of force *f* <sup>→</sup> (*xk* ) on the movement *h(s).* Using the same considerations, for *N* forces *f* → <sup>1</sup>, *f* → <sup>2</sup>,⋯, *f* → *<sup>N</sup>* that act independently in *N* points of abscissa *x*1, *x*2, ..., *xN* in the direction perpendicular to the string, the string movements will be obtained as *h*1, *h*2, ..., *h <sup>N</sup>* . Therefore the movement associated with an arbitrary point of the string of abscissa *s* is described by the relation below

$$h(\mathbf{s}) = \mathbf{g}(\mathbf{s}, \mathbf{x}\_1)f(\mathbf{x}\_1) + \mathbf{g}(\mathbf{s}, \mathbf{x}\_2)f(\mathbf{x}\_2) + \dots + \mathbf{g}(\mathbf{s}, \mathbf{x}\_k)f(\mathbf{x}\_k) = \sum\_{l=1}^{N} \mathbf{g}(\mathbf{s}, \mathbf{x}\_k)f(\mathbf{x}\_k) \tag{2}$$

When a force is distributed continuously along the entire string, the movement of the point *s* of the string will be given by

$$h(\mathbf{s}) = \int\_0 \mathbf{g}(\mathbf{s}, \mathbf{x}) f(\mathbf{x}) d\mathbf{x} \tag{3}$$

where *l* represents the length of the string.

**2. Mathematical background of signal deconvolution**

examples, the examples below should be noted.

142 Fourier Transform - Signal Processing and Physical Sciences

where *S* represents the surface of the body.

Figure 1.

),( <sup>11</sup> *hx*

structures also deal with ill-posed problems: see [3].

),( <sup>33</sup> *hx*

*kf* 

> 3*f*

> > force *f* →

2*f* 

**Figure 1.** Physical model of the vibrating string

),( <sup>22</sup> *hx*

that in the point of abscissa *xk*,

after speed is received from the dispersion curve analysis [2].

As mentioned in the introduction, ill-posed problems from a mathematical point of view have many applications in physics and technologies [1]. In addition to the abovementioned

Solving the Cauchy problem for the Laplace equation, *ΔU* =0 has a direct application in biophysics as in [2]. The problem consists in determining the biopotential distribution within the body denoted by *U*, when the body surface potential values are known. The phenomenon

The determination of radioactive substances in the body, as in [2], and protein crystallography

The same formalism is used in quantum mechanics to determine the particle scattering crosssection on different targets, as well as in plasma physics in the case of the electron distribution

An intuitive way of grasping an ill-posed problem can be modelled by the movement of the vibrating string when many forces are acting perpendicularly on the string, as represented in

),( <sup>44</sup> *hx*

In terms of the mathematical equation, the phenomenon described above has a correspondent in physics spectroscopy used in the study of nanomaterials. In the first instance, it is considered

),( *NN* -- <sup>22</sup> *hx*

*hx NN* ),( ),( *kk hx*

*<sup>N</sup>* -<sup>2</sup> *f* 

),( *NN* -- <sup>11</sup> *hx <sup>N</sup>*-<sup>1</sup> *f* 

*<sup>k</sup>* acts perpendicularly to the direction of the string. The

4*f* 

<sup>∂</sup>*<sup>n</sup>* <sup>|</sup>

*<sup>S</sup>* =0,

is modelled by the Laplace equation, and the Cauchy conditions are *<sup>U</sup>* <sup>|</sup>*<sup>S</sup>* <sup>=</sup> *<sup>f</sup>* (*S*) and ∂*<sup>U</sup>*

The function *f* is the density of force, which means the force per unit length, and *f(x)dx* represents the force that acts on the arc element *dx*. The function *g* is called the influence function because it shows the degree of influence of the distribution force *f* on displacement *h*.

The equation (3) is named the Fredholm integral equation of the first kind, and it is a particular case of the integral equation,

$$h(\mathbf{s})f(\mathbf{s}) + \bigwedge\_{a}^{b} \mathbf{g}(\mathbf{s}, \mathbf{x})f(\mathbf{x})d\mathbf{x} = h(\mathbf{s}), \quad \mathbf{c} \le \mathbf{x} \le d \tag{4}$$

where *l*, *g* and *h* are continuous known functions. If function *l* is null, then equation (4) represents an integral equation of the first kind. If function *l* has no zero on *c*, *d* then the equation (4) is of the second kind, while if *l* has some zeros on [*c,d*] then the equation (4) is of the third kind.

Although the aim of this chapter is signal deconvolution using the Fourier transform, it is important to mentionthe other two methods used to solve equation (4) when *l* ≡0, that is, the regularization method and the spline approach.

Hadamard stated that a problem is well posed if it has a unique solution, and the solution depends continuously on the data [4]. Any problem that is not a well-posed problem is an illposed one. The Fredholm equation of the first kind is ill posed because small changes in the data generate huge modification of the unknown function.

#### **2.1. Regularization method**

This method consists in the replacement of the ill-posed problem (4) with *l* ≡0 by a well-posed problem, and there are many scientific papers that develop different types of regularization method depending on kernel type and other specific needs. Below we describe the Tikhonov regularization method applied to the equation (4) with *l* ≡0. Let *X* and *Y* be Hilbert spaces and ⋅ be the norm on Hilbert space. If the kernel *g* is smooth, the operator *G* : *X* →*Y*

$$(Gf')(\mathbf{s}) = \bigwedge\_{a}^{b} \mathbf{g}(\mathbf{s}, \mathbf{x}) f(\mathbf{x}) d\mathbf{x} \tag{5}$$

is linear. Then equation (4) with *l* ≡0 becomes

$$
\delta Gf = h \tag{6}
$$

The regularization method consists in the determination of the approximate solution of the equation (4) of the first kind as a minimization of the following functional

$$\left\|\Phi\_a(f) = \left\|G \cdot h\right\|^2 + \alpha \left\|Lf\right\|^2, \,\forall f \in X\tag{7}$$

The value *α* >0 represents the regularization parameter and *L* is a linear operator defined below

$$Lf = a\_0(f - \hat{f}) + a\_1f' + a\_2f' \tag{8}$$

where *ai* has the value 0 or 1; and *f* ' and *f* '' are the first and the second derivative of *f*. Function *f* ^ represents a trial solution for equation (4) with *l* ≡0. The regularization order for the operator *L* is the same as the derivability order of *f.* The regularization parameter should be chosen carefully, because a good minimum for the functional (7) does not always lead to an adequate solution for equation (4) with *l* ≡0 as in [4]. The discrimination procedure of the equation (4) with *l* ≡0 and functional (7) depends on the specifics of each type of problem such as domain, type of kernel, etc., but this is not the subject of this chapter: see [4-6].

Some disadvantages of the regularization method, which is an iterative method, are the fact that it is very sensitive to the noise present in the experimental function and is time consuming. If the kernel g of the equation (4) with *l* ≡0 has a delayed argument then the equation is called a convolution equation, and is widely applied in physical spectroscopy. The general form of a convolution equation is given by

$$h(s) = \int\_{s}^{s} \mathbf{g}(s - \mathbf{x}) f(\mathbf{x}) d\mathbf{x} \tag{9}$$

After the change of variable *x=t-s* it is found that (9) is equivalent by the equation

$$h(s) = \bigcap\_{s}^{a} \mathbf{g}(t) f(t - s) dt \tag{10}$$

#### **2.2. Spline technique**

Hadamard stated that a problem is well posed if it has a unique solution, and the solution depends continuously on the data [4]. Any problem that is not a well-posed problem is an illposed one. The Fredholm equation of the first kind is ill posed because small changes in the

This method consists in the replacement of the ill-posed problem (4) with *l* ≡0 by a well-posed problem, and there are many scientific papers that develop different types of regularization method depending on kernel type and other specific needs. Below we describe the Tikhonov regularization method applied to the equation (4) with *l* ≡0. Let *X* and *Y* be Hilbert spaces and

The regularization method consists in the determination of the approximate solution of the

2 2

The value *α* >0 represents the regularization parameter and *L* is a linear operator defined below

 represents a trial solution for equation (4) with *l* ≡0. The regularization order for the operator *L* is the same as the derivability order of *f.* The regularization parameter should be chosen carefully, because a good minimum for the functional (7) does not always lead to an adequate solution for equation (4) with *l* ≡0 as in [4]. The discrimination procedure of the equation (4) with *l* ≡0 and functional (7) depends on the specifics of each type of problem such as domain,

Some disadvantages of the regularization method, which is an iterative method, are the fact that it is very sensitive to the noise present in the experimental function and is time consuming.

' ''

*Gf s g s x f x dx* <sup>=</sup> ò (5)

() - , *<sup>a</sup>* F = +a " Î *f G h Lf f X* (7)

0 12 <sup>ˆ</sup> *Lf a f f a f a f* = -+ + ( ) (8)

are the first and the second derivative of *f*. Function

*Gf h* = (6)

⋅ be the norm on Hilbert space. If the kernel *g* is smooth, the operator *G* : *X* →*Y*

( )() (, ) () *b*

*a*

equation (4) of the first kind as a minimization of the following functional

and *f* ''

type of kernel, etc., but this is not the subject of this chapter: see [4-6].

data generate huge modification of the unknown function.

is linear. Then equation (4) with *l* ≡0 becomes

has the value 0 or 1; and *f* '

**2.1. Regularization method**

144 Fourier Transform - Signal Processing and Physical Sciences

where *ai*

*f* ^ Spline functions for signal deconvolution technique help eliminate the drawbacks mentioned above [7]. The advantage of the method proposed in [7] lies in the fact that Beniaminy's method is a one-step method. In this case, the true sample function *f* is represented as a piecewise cubic spline function, and after the substitution of it into equation (10), the experimental function *h* becomes a piecewise cubic spline function with the same knots but different coefficients. The connection between the coefficients of functions *h* and *f* are given by the moments of instru‐ mental function *g*. Thus, if function *f* has the form

$$\int f(t) = \sum\_{k=1}^{n-1} s\_k(t) \tag{11}$$

where

$$s\_k(t) = \begin{cases} 0 & \text{if } \quad t < \xi\_k \\ a\_k t^3 + b\_k t^2 + c\_k t + d\_k & \text{if } \quad \xi\_k \le t \le \xi\_{k+1} \\ 0 & \text{if } \quad \xi\_{k+1} < t \end{cases} \tag{12}$$

with *ξ<sup>k</sup>* , *k* =1 ¯ , *n* −1 are the knots and *ak*, *bk*, *ck* and *dk* are the coefficients of the spline function *f*. They are chosen such that the function together with its first two derivatives is continuous. Replacing (12) in (11), the experimental function has the form

$$h(\mathbf{s}) = \sum\_{k=1}^{n-1} \left( A\_k t^3 + B\_k t^2 + C\_k t + D\_k \right) \tag{13}$$

where

$$\begin{array}{rcl} A\_k & = & a\_k M\_0 \\ B\_k & = & b\_k M\_0 - 3a\_k M\_1 \\ C\_k & = & 3a\_k M\_2 - 2b\_k M\_1 + c\_k M\_0 \\ D\_k & = & -a\_k M\_3 + b\_k M\_2 - c\_k M\_1 + d\_k M\_0 \end{array} \tag{14}$$

and *Mk* = *∫* −*∞ ∞ t <sup>k</sup> g*(*t*)*dt* represents the moment of order *k*. Beniaminy considered that experimental function *h* given by (14) is a cubic spline function. In [8] it is shown that that function *h* is not a spline function due to the lack of the continuity in the first two derivatives of *h.* However, the algorithm from [7] gives good results, but the quality of the true sample function depends on how wide the instrumental function is. In order to obtain the true sample function *f* given by (12) and (13), we calculate spline coefficients of the experimental function and using these values and (14) obtain the coefficients *ak*, *bk*, *ck* and *dk.*

#### **2.3. Solving the convolution equation using Fourier transform**

Take the functions *h*, *f*, and *g* whose Fourier transform is given by the functions *H*, *F* and *G*. By applying the Fourier transform operator on both members of equation (10) we obtain

$$\int\_{-\alpha}^{\ast \alpha} h(s) \exp(-2\pi i \text{vs}) ds = \int\_{-\alpha}^{\ast \alpha} \left[ \int\_{-\alpha}^{\ast \alpha} \mathbf{g}(\tau) f(s - \tau) d\tau \exp(-2\pi i \text{vs}) \right] ds \tag{15}$$

By changing the order of integration, the Fourier transform of the signal *h* is expressed by the relation,

$$H(\mathbf{v}) = \int\_{-\alpha}^{\alpha} \mathbf{g}(\mathbf{r}) \left[ \int\_{-\alpha}^{\ast \alpha} f(s - \mathbf{r}) \exp(-2\pi i \mathbf{v} \, s) \, ds \right] d\mathbf{r} \tag{16}$$

Using the substitution *σ* =*s* −*ν*, the quantity between square brackets from the previous relation becomes

$$\int\_{-\alpha}^{+\alpha} f(\sigma) \exp\left[-2\pi i \mathbf{v} \left(\sigma + \mathfrak{r}\right)\right] d\sigma = \exp\left(-2\pi i \mathbf{v} \mathfrak{r}\right) \int\_{-\alpha}^{+\alpha} f(\sigma) \exp(-2\pi i \mathbf{v} \sigma) d\sigma = \exp(-2\pi i \mathbf{v} \mathfrak{r}) F(\mathbf{u}) \tag{17}$$

In this context the relation (16) becomes

$$H(\mathbf{v}) = \int\_{-\alpha}^{+\alpha} \mathbf{g}(\tau) \exp(-2\pi i \mathbf{v}\tau) F(\mathbf{v}) \,d\tau = F(\mathbf{v})G(\mathbf{v}) \tag{18}$$

*i s ds d*

( ) ( ) ( ) exp(2 ) (16)

 

> *i* )*F*(

) (15)

) (18)

(17)

The relation (18) is known as the convolution theorem. If direct and inverse Fourier transform operators and convolution product are respectively denoted by TF, TF-1 and \*, then the relation (18) is written symbolically as Using the substitution *s* , the quantity between square brackets from the previous relation becomes 

*<sup>h</sup> <sup>s</sup> <sup>i</sup> <sup>s</sup> ds <sup>g</sup> <sup>f</sup> <sup>s</sup> <sup>d</sup> <sup>i</sup> <sup>s</sup> ds*

By changing the order of integration, the Fourier transform of the signal *h* is expressed by the

 ) ( ) exp(2 

 

) (

*f s*

 exp 2*i*

> ) exp(2

and instrumental contribution *g* measured on a gold foil, are presented in Figure 2.

 

( ) exp(2

*H g*

*H* () *g*(

 

 )exp 2*i d*

In this context the relation (16) becomes

$$TF(h) = TF(f) \\ TF(\mathbf{g}) = F \ G = TF(f \ast \mathbf{g})$$

 *f* ( )exp(2*i* )*d*exp(2

> *i* )*F*( ) *d F*( )*G*(

and

relation,

*f* (

written symbolically as

(14)

0

*kk k*

= -

*B bM aM*

*k k*

=

146 Fourier Transform - Signal Processing and Physical Sciences

values and (14) obtain the coefficients *ak*, *bk*, *ck* and *dk.*

+¥ +¥ +¥



+¥


In this context the relation (16) becomes

+¥ +¥


**2.3. Solving the convolution equation using Fourier transform**

and *Mk* = *∫*

relation,

becomes

−*∞*

*∞*

*A aM*

0 1

*k kkkk*

*D aM bM cM dM*

=- + - +

3 3 2

*k k kk*

*C aM bM cM*

= -+

2 10

function *h* given by (14) is a cubic spline function. In [8] it is shown that that function *h* is not a spline function due to the lack of the continuity in the first two derivatives of *h.* However, the algorithm from [7] gives good results, but the quality of the true sample function depends on how wide the instrumental function is. In order to obtain the true sample function *f* given by (12) and (13), we calculate spline coefficients of the experimental function and using these

Take the functions *h*, *f*, and *g* whose Fourier transform is given by the functions *H*, *F* and *G*. By


By changing the order of integration, the Fourier transform of the signal *h* is expressed by the

é ù n = t -t - pn t ê ú

Using the substitution *σ* =*s* −*ν*, the quantity between square brackets from the previous relation

s é- p n s + t ù s = - p nt s - p ns s = - p nt u ò ò ë û (17)

é ù

ë û <sup>ò</sup> ò ò (15)

ë û ò ò (16)

n = t - p nt n t = n n ò (18)

applying the Fourier transform operator on both members of equation (10) we obtain

*h s i s ds g f s d i s ds* ( )exp( 2 ) ( ) ( ) exp( 2 )

*H g fs* ( ) ( ) ( )exp( 2 ) *i s ds d*

*f i d i f i d iF* ( )exp 2 ( ) exp 2 ( )exp( 2 ) exp( 2 ) ( ) ( ) +¥ +¥

*H g iF d FG* ( ) ( )exp( 2 ) ( ) ( ) ( )

321 0

*t <sup>k</sup> g*(*t*)*dt* represents the moment of order *k*. Beniaminy considered that experimental

$$TF^{-1}(F\,G) = h = f \ast \mathbf{g}$$

In this way, the process of the inverse Fourier transform applied to function *F* determines *f* signal. In X-ray diffraction theory this is known as the Stokes method. *TF*(*h*) *TF*( *f* )*TF*(*g*) *F G TF*( *f g*) and *TF F G h f g* ( ) <sup>1</sup>

Experimental signals *h*, coded by (1), (3), (5) and (6) for a set of supported gold catalyst (Au/ SiO2), and instrumental contribution *g* measured on a gold foil, are presented in Figure 2. In this way, the process of the inverse Fourier transform applied to function *F* determines *f* signal. In X-ray diffraction theory this is known as the Stokes method. Experimental signals *h*, coded by (1), (3), (5) and (6) for a set of supported gold catalyst (Au/SiO2),

Figure 2 The experimental relative intensities *h* of the supported gold catalysts and instrumental function *g* **Figure 2.** The experimental relative intensities *h* of the supported gold catalysts and instrumental function *g*

**3. Why is the technique of deconvolution used in nanomaterials science?** 

#### In the scientific literature we can see many authors display serious confusion about the concept of **3. Why is the technique of deconvolution used in nanomaterials science?**

deconvolution. Often, when they decompose the experimental signal *h* according to certain specific

criteria, some say that it has achieved the deconvolution of the initial signal. This fact may be accepted only if the instrumental function g from equation (11) is described by the Dirac distribution. In the scientific literature we can see many authors display serious confusion about the concept of deconvolution. Often, when they decompose the experimental signal *h* according to certain specific criteria, some say that it has achieved the deconvolution of the initial signal. This fact may be accepted only if the instrumental function g from equation (11) is described by the Dirac distribution. Only in this case is the true sample function *f* identical to the experimental signal *h*. Unfortunately, no instrumental function of any measuring device can be described by the Dirac distribution.

It is well known that the macroscopic physical properties of various materials depend directly on their density of states (DS). The DS is directly linked to crystallographic properties. For physical systems that belong to the long order class, moving the crystallographic lattice in the whole real space will reproduce the whole structure. The nanostructured materials, which belong to the short-range class, are obtained by moving the lattice in the three crystallographic directions at the limited distances, generating crystallites whose size is no greater than a few hundred angstroms. In this case, the DS is drastically modified in comparison with the previous class of materials. From a physical point of view the DS is closely related to the nanomaterials' dimensionality, so crystallite size gives direct information about new topolog‐ ical properties. It can emphasize that amorphous, disordered or weak crystalline materials can have new bonding and anti-bonding options. The systems consisting of nanoparticles whose dimensions do not exceed 50 Å have the majority of atoms practically situated on the surface for the most part. Additionally, the behaviour of crystallites whose size is between 50 Å and 300 Å is described on the basis of quantum mechanics to explain the advanced properties of the tunnelling effect. All these reasons lead to the search for an adequate method to determine reliable information such as effective particle size, microstrains of lattice, and particle distri‐ bution function. This information is obtained by Fourier deconvolution of the instrumental and experimental X-ray line profiles (XRLP) approximated by Gauss, Cauchy and Voigt distributions and generalized by Fermi function (GFF) as in [9]. The powder reflection broadening of the nanomaterials is normally caused by small size, crystallites and distortions within crystallites due to dislocation configurations. It is the most valuable and cheapest technique for the structural determination of crystalline nanomaterials.

Generally speaking, in X-ray diffraction on powder, the most accurate and reliable analysis of the signals is given by the convolution equation (10) where *h*, *g* and *f* are experimental data, instrumental contribution of setup experimental spectrum, and true sample function as a solution of equation (10), respectively.

**Figure 3.** Numerical solution of the deconvolution equation (19) determined by an algebraic discretization

Let us consider the experimental signals of (111) X-ray line profile of supported nickel catalyst, and the instrumental function given by nickel foil obtained by a synchrotron radiation setup at 201 points with a constant step of 0.040 in 2θ variables, as shown in Figure 3.

The convolution equation (9) can be approximated in different ways, but the simplest approx‐ imation is given by following the algebraic system

$$h(\mathbf{x}\_i) = \sum\_{j=-201}^{201} \mathbf{g}(\mathbf{x}\_i - \mathbf{s}\_j) f(\mathbf{s}\_j) \Delta \mathbf{s}\_j \dots \mathbf{i} = \text{l,201} \tag{19}$$

where *Δsj* is a constant step in 2θ variables. It turns out that the roots *f(sj )* of system (19) do not lead to a smooth signal, but yield a curve which makes for enhanced oscillations. Its behaviour is given by *f* signal in Figure 3. This result is given by a computer code written in Maple 11 language, a sequence of which is presented in Appendix 1. From a physical point of view, this type of solution is impracticable because the crystallite size in nanostructured systems is contained in the tails of XRLP. Therefore, the lobes of the XRLP must be sufficiently smooth. As shown in the inset of Figure 3, this condition is not met.. It would be possible to improve the quality of signal *f* trying to extend the definition interval for signal g. Thus we will approximate the unbounded integral on a bounded interval, but one that is sufficiently large.

This depends on the performances of the computer system and on the algorithm developed for solving inhomogeneous systems of linear equations with sizes of at least several thousand.

## **4. Distributions frequently used in physics and chemical signal deconvolution applied in nanomaterials science**

It is known that, from a mathematical point of view, the XRLP are described by the symmetric or asymmetric distributions. As in [10,11] a large variety of functions for analysis of XRLP, such as Voigt (V), pseudo-Voigt (pV) and Pearson VII (P7), are proposed.

#### **4.1. Gauss distribution**

whole real space will reproduce the whole structure. The nanostructured materials, which belong to the short-range class, are obtained by moving the lattice in the three crystallographic directions at the limited distances, generating crystallites whose size is no greater than a few hundred angstroms. In this case, the DS is drastically modified in comparison with the previous class of materials. From a physical point of view the DS is closely related to the nanomaterials' dimensionality, so crystallite size gives direct information about new topolog‐ ical properties. It can emphasize that amorphous, disordered or weak crystalline materials can have new bonding and anti-bonding options. The systems consisting of nanoparticles whose dimensions do not exceed 50 Å have the majority of atoms practically situated on the surface for the most part. Additionally, the behaviour of crystallites whose size is between 50 Å and 300 Å is described on the basis of quantum mechanics to explain the advanced properties of the tunnelling effect. All these reasons lead to the search for an adequate method to determine reliable information such as effective particle size, microstrains of lattice, and particle distri‐ bution function. This information is obtained by Fourier deconvolution of the instrumental and experimental X-ray line profiles (XRLP) approximated by Gauss, Cauchy and Voigt distributions and generalized by Fermi function (GFF) as in [9]. The powder reflection broadening of the nanomaterials is normally caused by small size, crystallites and distortions within crystallites due to dislocation configurations. It is the most valuable and cheapest

technique for the structural determination of crystalline nanomaterials.

solution of equation (10), respectively.

148 Fourier Transform - Signal Processing and Physical Sciences

Generally speaking, in X-ray diffraction on powder, the most accurate and reliable analysis of the signals is given by the convolution equation (10) where *h*, *g* and *f* are experimental data, instrumental contribution of setup experimental spectrum, and true sample function as a

**Figure 3.** Numerical solution of the deconvolution equation (19) determined by an algebraic discretization

Many results such as the propagation of uncertainties and the least square method can be derived analytically in explicit form when the relevant variables are normally distributed. Gauss distribution is defined by mathematical relation

$$I\_G = \frac{I\_{0G}}{\sqrt{\pi \gamma\_G}} \exp\left[-\left(\frac{x-a}{\gamma\_G}\right)^2\right] \tag{20}$$

where *I0G*, *a* and γG are the profile area, gravitational centre measured in 2*θ* variable, and broadening of the XRLP, respectively. The *n*th moment, *n*=0,1 is given by relations

$$\mu\_{0G} = \bigcap\_{\rightsquigarrow} I\_G(\mathbf{x})d\mathbf{x} = I\_{0G}, \qquad \mu\_{1G} = a$$

The integral width *δG* and full width at half maximum FWHMG are given by relations

$$
\delta\_G = \sqrt{\pi} \gamma\_G \quad \text{and} \quad FWHM\_G = 2\sqrt{\ln 2} \gamma\_G
$$

If both signals *h* and *g* are described by Gaussian distributions and take into account the relationship (18), the full width and FWHM of the true sample function are expressed by the relations

$$\gamma\_{G,f} = \sqrt{\gamma\_{G,h}^2 - \gamma\_{G,g}^2} \quad FFWHM\_{G,f} = 2\sqrt{\ln 2} \chi\_{G,f} \tag{21}$$

#### **4.2. Cauchy distribution**

The Cauchy distribution, also called the Lorentzian distribution, is a continuous distribution that describes population distribution of electron levels with multiple applications in physical spectroscopy. Its analytical expression is given by relation

$$I\_C = \frac{I\_{0C}}{\pi} \frac{\gamma\_C}{\gamma\_C^2 + (\chi - a)^2} \tag{22}$$

where *I*0*C*, *a* and *γC* are profile surface, gravitational centre and broadening of the XRLP, respectively. The *n*th moment *n*=0,1 is given by relations

$$\mu\_{0C} = \bigcap\_{\sim \mathcal{C}} I\_{\mathcal{C}}(\mathbf{x})d\mathbf{x} = I\_{\text{0}C} \quad \text{and} \quad \mu\_{\text{1C}} = a$$

The integral widths *δG* and full width at half maximum *FWHMC* are given by relations

$$
\delta\_C = \pi \gamma\_C \quad \text{and} \quad FWHM\_C = 2\gamma\_C
$$

The deconvolution of two signals h and g determined by Cauchy distributions is also a Cauchy distribution whose full width *δC,f* and *FWHMC,f* are given by relations

$$
\delta \delta\_{C,f} = \delta\_{C,h} - \delta\_{C,g} \quad \text{and} \quad FWHM\_{C,f} = \mathcal{Z}\delta\_{C,f} \tag{23}
$$

#### **4.3. Generalized Fermi function**

<sup>0</sup> 0 1 () , *GG G G I x dx I a*

*δ<sup>G</sup>* = *πγ<sup>G</sup>* and *FWH MG* =2 ln2*γ<sup>G</sup>*

If both signals *h* and *g* are described by Gaussian distributions and take into account the relationship (18), the full width and FWHM of the true sample function are expressed by the

The Cauchy distribution, also called the Lorentzian distribution, is a continuous distribution that describes population distribution of electron levels with multiple applications in physical

> 2 2 ( ) *C C*

where *I*0*C*, *a* and *γC* are profile surface, gravitational centre and broadening of the XRLP,

0 01 ( ) and *CC C C I x dx I a*

and 2 *C C FWHMC C* d = pg = g

The deconvolution of two signals h and g determined by Cauchy distributions is also a Cauchy

, ,, , , and 2 *C f Ch Cg FWHMCf Cf* d =d -d = d (23)

m= = m= ò

The integral widths *δG* and full width at half maximum *FWHMC* are given by relations

0

*C <sup>I</sup> <sup>I</sup> x a*

*C*

+¥


distribution whose full width *δC,f* and *FWHMC,f* are given by relations

, ,, , , 2 ln 2 *G f Gh Gg FWHMG f G f* g = g -g = g (21)

<sup>g</sup> <sup>=</sup> pg + - (22)

m= = m= ò

The integral width *δG* and full width at half maximum FWHMG are given by relations

+¥

150 Fourier Transform - Signal Processing and Physical Sciences


2 2

spectroscopy. Its analytical expression is given by relation

respectively. The *n*th moment *n*=0,1 is given by relations

relations

**4.2. Cauchy distribution**

Although extensive research over the past few decades has made progress in XRLP global approximations, their complete analytical properties have not been reported in the literature. Unfortunately, most of them have complicated forms, and they are not easy to handle mathematically. Recently, as in [9,11], a simple function with a minimal number of parameters named the generalized Fermi function (GFF), suitable for minimization and with remarkable analytical properties, was presented from a purely phenomenological point of view. It is given by the relationship,

$$h(s) = \frac{A}{e^{-a(s-\varepsilon)} + e^{b(s-\varepsilon)}}\tag{24}$$

where *A, a, b, c* are unknown parameters. The values *A, c* describe the amplitude and the position of the peak, and *a, b* control its shape. If *b=0*, the *h* function reproduces the Fermi-Dirac electronic energy distribution. The GFF has remarkable mathematical properties, with direct use in determining the moments, the integral width, and the Fourier transform of the XRLP, as well as the true sample function. Here we give its properties without proofs.

**i.** By setting

$$\mathbf{s}" = \mathbf{s} - \mathbf{c} \quad \mathbf{p} = (a+b)/2 \quad q = (a-b)/2$$

we obtain

$$h(\mathbf{s'}) = \frac{A}{2} \left( \frac{\cosh q \mathbf{s'} + \sinh q \mathbf{s'}}{\cosh \mathbf{p} \, \mathbf{s'}} \right) \tag{25}$$

**ii.** the limit of *h* function for infinite arguments is finite, so lim*h* (*s*')=0 when *s*'→ ± *∞* ;

**iii.** the zero, first and second order moments (*µ0, µ1, µ2*) of the *h* function are given by the relations

$$\mu\_0 = \frac{\pi \mathcal{A}}{2\rho \cos\frac{\pi q}{2\rho}}, \quad \mu\_1 = \frac{\pi}{2\rho} \tan\frac{\pi q}{2\rho}, \quad \mu\_2 = \left(\frac{\pi}{2\rho}\right)^2 \left(\frac{1}{\cos^2\frac{\pi q}{2\rho}} + \tan^2\frac{\pi q}{2\rho}\right).$$

**iv.** the integral width *δh(a, b)* of the *h* function has the following form

$$\delta\_h(a,b) = \frac{\pi}{\left(a^a b^b\right)^{1/(a\*b)} \cos\left(\frac{\pi}{2} \frac{a-b}{a\*b}\right)}\tag{26}$$

#### **v.** the Fourier transform of the *h* function is given by the relationship

$$H(L) = \frac{A}{2} \int\_{-\alpha}^{\alpha \epsilon} \frac{\cosh qs^\* + \sinh qs^\*}{\cosh \rho s^\*} e^{-2\pi s^\* L} ds^\* = \frac{\pi A}{2\rho \left| \cos \left( \frac{\pi q}{2\rho} + i \frac{\pi^2 L}{\rho} \right) \right|^2} \cos \left( \frac{\pi q}{2\rho} - i \frac{\pi^2 L}{\rho} \right) \tag{27}$$

**vi.** if we consider the functions *f* and *g* defined by equation (25), by their deconvolution we can compute the *|F(L)|* function, which is used in Warren and Averbach's analysis in [12]. Therefore, the magnitude of *F(L)* function has the following form:

$$\left| F(L) \right| = \frac{A\_h \rho\_g}{A\_g \rho\_h} \sqrt{\frac{\cos^2 \alpha + \sinh^2 \beta L}{\cos^2 \gamma + \sinh^2 \delta L}},\tag{28}$$

where the arguments of trigonometric and hyperbolic functions are expressed by

$$\alpha = \frac{\pi q\_g}{2\rho\_g}, \quad \beta = \frac{\pi^2}{\rho\_g}, \quad \gamma = \frac{\pi q\_h}{2\rho\_h}, \quad \delta = \frac{\pi^2}{\rho\_h}$$

The subscripts *g* and *h* refer to the instrumental and experimental XRLP. Taking into account the convolution theorem, the true sample function *f* is given by the relationship

$$f(\mathbf{s}) = \frac{A\_h \mathbf{p}\_g}{A\_g \mathbf{p}\_h} \prod\_{\approx 0}^{\approx \approx} \frac{\cos\left(\frac{\pi q\_g}{2\rho\_g} + i \frac{\pi^2 L}{\rho\_g}\right)}{\cos\left(\frac{\pi q\_h}{2\rho\_h} + i \frac{\pi^2 L}{\rho\_h}\right)} \exp(2\pi i L \mathbf{s}) \,\mathrm{d}s$$

The last integral cannot be accurately resolved. In order to do so we have to consider some arguments. The Fourier transform of *f* is the *F* function, given by the relations

$$F(L) = \left| F(L) \right| \exp(i\theta(L)), \quad \theta(L) = \arctan \frac{\mathfrak{T}(F(L))}{\mathfrak{R}\left(F(L)\right)}$$

where *θ* means the angle function, and ℜ(*F* ) and ℑ(*F*) are the real and imaginary parts of the complex function *F*, respectively. The arguments *α, β, γ* and *δ* from equation (28) depend only on the asymmetry parameters *a* and *b* of the *g* and *f* functions. If the XRLP asymmetry is not very large (i.e., *a* and *b* parameters are close enough as values) the cos<sup>2</sup> *α* ≈1, cos<sup>2</sup> *γ* ≈1 approxi‐ mations are reliable. Therefore, we obtain ℑ(*F*)< <ℜ(*F* ), *θ(L) ≈ 0* and the magnitude of the Fourier transform for the true XRLP sample can be expressed as

Experimental Data Deconvolution Based on Fourier Transform Applied in Nanomaterial Structure http://dx.doi.org/10.5772/59667 153

$$\left| F(L) \right| = \frac{A\_h \mathfrak{p}\_g}{A\_g \mathfrak{p}\_h} \frac{\cosh \frac{\pi^2 L}{\mathfrak{p}\_g}}{\cosh \frac{\pi^2 L}{\mathfrak{p}\_h}} \tag{29}$$

(vii) if we consider the previous approximation, the true XRLP sample is given by an inverse Fourier transform of the *F* function, and consequently we have

$$f(\mathbf{s'}) = \frac{2A\_h \mathbf{p}\_g}{\pi A\_g} \frac{\cos \frac{\pi \rho\_h}{2\rho\_g} \cosh \rho\_h \mathbf{s'}}{\cosh 2\rho\_h \mathbf{s'} + \cos \frac{\pi \rho\_h}{\rho\_g}} \tag{30}$$

(viii) the integral width of the true XRLP sample can be expressed by the *δ<sup>f</sup>* function

$$\delta\_{\mathcal{I}\_{\mathcal{I}}}(\boldsymbol{\wp}\_{h}, \boldsymbol{\wp}\_{\mathcal{g}}) = \frac{\pi}{2\rho\_{h}\cos\frac{\pi\rho\_{h}}{2\rho\_{\mathcal{J}}}} \Big(\cos\frac{\pi\rho\_{h}}{\rho\_{\mathcal{J}}} + \mathbb{I}\Big) \tag{31}$$

#### **4.4. Voigt distribution applied in X-ray line profile analysis**

**v.** the Fourier transform of the *h* function is given by the relationship

2 '

2 cosh ' 2 cos

+¥

152 Fourier Transform - Signal Processing and Physical Sciences

cosh ' sinh ' ( ) ' cos

*A qs qs <sup>A</sup> H L e ds <sup>i</sup> <sup>s</sup> <sup>i</sup>*

<sup>+</sup> <sup>=</sup> <sup>+</sup>

where the arguments of trigonometric and hyperbolic functions are expressed by

the convolution theorem, the true sample function *f* is given by the relationship

*h g*

*h g g h*

r

r

cos sinh ( ) , cos sinh

,, , 2 2 *g h*

<sup>p</sup>*<sup>q</sup>* pp p *<sup>q</sup>* a= b= g= d= rr rr

The subscripts *g* and *h* refer to the instrumental and experimental XRLP. Taking into account

( ) ( )

*q L*

2 2

<sup>p</sup> <sup>p</sup> +¥ r r <sup>p</sup> <sup>p</sup> -¥ r r

( ) exp(2 )

*f s iLs ds*

The last integral cannot be accurately resolved. In order to do so we have to consider some

( ) ( ) ( ) exp( ( )), ( ) arctan ( )

where *θ* means the angle function, and ℜ(*F* ) and ℑ(*F* ) are the real and imaginary parts of the complex function *F*, respectively. The arguments *α, β, γ* and *δ* from equation (28) depend only on the asymmetry parameters *a* and *b* of the *g* and *f* functions. If the XRLP asymmetry is not

mations are reliable. Therefore, we obtain ℑ(*F*)< <ℜ(*F* ), *θ(L) ≈ 0* and the magnitude of the

<sup>Á</sup> = q q=

*g g g h h h*

cos

r + = p <sup>r</sup> <sup>+</sup> ò

cos

*A i*

arguments. The Fourier transform of *f* is the *F* function, given by the relations

*FL FL i L L*

very large (i.e., *a* and *b* parameters are close enough as values) the cos<sup>2</sup>

Fourier transform for the true XRLP sample can be expressed as

*<sup>q</sup> <sup>L</sup> g h A i*

2 2

*<sup>A</sup> <sup>L</sup> F L A L*

r r <sup>p</sup> <sup>p</sup> -¥ r r + p <sup>=</sup> <sup>=</sup> - <sup>r</sup> r +

**vi.** if we consider the functions *f* and *g* defined by equation (25), by their deconvolution

in [12]. Therefore, the magnitude of *F(L)* function has the following form:

a

g

we can compute the *|F(L)|* function, which is used in Warren and Averbach's analysis

2 2 2 2

*g g hh*

( ) ( ) <sup>2</sup> 2

2

 b

 d

2 2

*s L q L q L*


ò (27)

2 2

( ) ( )

*F L*

*F L*

*α* ≈1, cos<sup>2</sup>

*γ* ≈1 approxi‐

Â

(28)

Before briefly describing the mathematical properties of the Voigt distribution, let us examine the physical concept underlying the approximation of the XRLP by Voigt distribution and the convolution process.

During decades of research, Warren and Averbach [12] introduced the X-ray diffraction concept for the mosaic structure model, in which the atoms are arranged in blocks, each block itself being an ideal crystal, but with adjacent blocks that do not accurately fit together. They considered that the XRLP h represents the convolution between the true sample f and the instrumental function g, produced by a well-annealed sample. The effective crystallite size *Deff* and lattice disorder parameter *<εhkl>* were analysed as a set of independent events in a likelihood concept. Based on Fourier convolution produced between f and g signals and the mosaic structural model, the analytical form of the Fourier transform for the true sample function was obtained. The normalized *F* was described as the product of two factors, F(s)(L) and F(*ε*) (L) , where variable *L* represents the distance perpendicular to the (hkl) reflection planes. The factor F(s)(L) describes the contribution of crystallite size and stocking fault probability, while the factor F(*ε*) (L) gives information about the microstrain of the lattice. The general form of the Fourier transform of the true sample for cubic lattices was given by relationships

$$F^{(s)}(L) = e^{-\frac{\|L\|}{D\_{\varepsilon\|}\langle kld\rangle}}, \qquad F^{(e)}(L) = e^{-\frac{2\pi^2\left\{e\_1^2\right\}\_{kld}h\_0^2L^2}{a^2}},\tag{32}$$

where *h*<sup>0</sup> <sup>2</sup> =*h* <sup>2</sup> + *k* <sup>2</sup> + *l* <sup>2</sup> . The general form of the true sample function *f* is given by inverse Fourier transform of *F(L)*

$$f\left(s\right) = \int\_{-\kappa}^{\kappa} e^{-\left\|f\right\|^2 - \gamma\left\|t\right\|} e^{2\pi itsL} dL = $$

$$= \sqrt{\frac{\pi}{\theta}} \exp\left[\frac{\gamma^2 - \left(2\pi s\right)^2}{4\theta}\right] \left|\Re\left(\operatorname{erfc}\left(\frac{\gamma - 2\pi ts}{2\sqrt{\theta}}\right)\right) \cos\frac{\pi\gamma s}{\theta} - \Im\left(\operatorname{erfc}\left(\frac{\gamma + 2\pi ts}{2\sqrt{\theta}}\right)\right) \sin\frac{\pi\gamma s}{\theta}\right| \tag{33}$$

where *<sup>s</sup>* =2( sin*<sup>θ</sup> <sup>λ</sup>* <sup>−</sup> sin*θ*<sup>0</sup> *<sup>λ</sup>* ), *erf* (*x*)= <sup>2</sup> *π ∫* 0 *x e* <sup>−</sup>*<sup>t</sup>* <sup>2</sup> *dt*, *erfc*(*x*)=1−*erf* (*x*) is the complementary error

function [13] and *β* = <sup>2</sup>*<sup>π</sup>* <sup>2</sup> *<sup>ε</sup><sup>L</sup>* 2 *hkl h*0 2 *<sup>a</sup>* <sup>2</sup> , *<sup>γ</sup>* <sup>=</sup> <sup>1</sup> *Deff* (*hkl*) . The last relation from the mathematical point of view represents a Voigt distribution. If we take into account the properties of the Gauss and Cauchy distributions, the Voigt distribution can be generalized by relation

$$V(\mathbf{x}, \boldsymbol{\gamma}\_G, \boldsymbol{\gamma}\_C) = \bigcap\_{\ast=0}^{\ast \ast \ast} I\_G(\mathbf{x}', \boldsymbol{\gamma}\_G, \mathbf{x}\_{0G}) I\_C(\mathbf{x} - \mathbf{x}', \boldsymbol{\gamma}\_C, \mathbf{x}\_{0C}) d\mathbf{x}' \tag{34}$$

Based on relation (18), its Fourier transform is given by *FT V* = *FT IG FG* ⋅ *FT IC FC* where

$$F\_G(L) = e^{-2\pi i \chi\_{0G} L\_\nu - \pi^2 \chi\_G^2 L^2} \quad \text{and} \quad F\_C(L) = e^{-2\pi i \chi\_{0C} L - 2\pi \chi\_C} \|\cdot\|$$

The analytical expression of the Voigt distribution is

$$V(\mathbf{x}, \boldsymbol{\gamma}\_G, \boldsymbol{\gamma}\_C) = FT^{-1} \left[ e^{-2\pi t (\boldsymbol{\chi}\_{0C} \cdot \mathbf{x}\_{0G})L} e^{-\pi^2 \boldsymbol{\gamma}\_G^2 L^2 - 2\pi \boldsymbol{\gamma}\_C |\boldsymbol{\omega}|} \right] \tag{35}$$

Explicit forms of experimental signal and true sample function normalized at *I*0*<sup>V</sup>* are given by relations [13]

$$\begin{split} V(\mathbf{x}, \mathbf{y}\_G, \mathbf{y}\_C) &= \frac{I\_{0V}}{\sqrt{\pi \chi\_G}} \exp\left[\frac{\chi\_C^2 - (\mathbf{x} - \mathbf{x}\_{0C} - \mathbf{x}\_{0G})^2}{\chi\_G^2}\right]. \\ &\cdot \left\{ \Re\left[\text{erfc}\left(\frac{\chi\_C - i(\mathbf{x} - \mathbf{x}\_{0C} - \mathbf{x}\_{0G})}{\chi\_G}\right)\right] \cos\frac{2\chi\_C(\mathbf{x} - \mathbf{x}\_{0C} - \mathbf{x}\_{0G})}{\chi\_G^2} - \\ &- \Im\left[\text{erfc}\left(\frac{\chi\_C + i(\mathbf{x} - \mathbf{x}\_{0C} - \mathbf{x}\_{0G})}{\chi\_G}\right)\right] \sin\frac{2\chi\_C(\mathbf{x} - \mathbf{x}\_{0C} - \mathbf{x}\_{0G})}{\chi\_G^2} \right\} \end{split} \tag{36}$$

Experimental Data Deconvolution Based on Fourier Transform Applied in Nanomaterial Structure http://dx.doi.org/10.5772/59667 155

and

(33)

where *h*<sup>0</sup>

transform of *F(L)*

where *<sup>s</sup>* =2( sin*<sup>θ</sup>*

relations [13]

*G C*

function [13] and *β* =

<sup>2</sup> =*h* <sup>2</sup> + *k* <sup>2</sup> + *l* <sup>2</sup>

154 Fourier Transform - Signal Processing and Physical Sciences

*<sup>λ</sup>* <sup>−</sup>

. The general form of the true sample function *f* is given by inverse Fourier

*dt*, *erfc*(*x*)=1−*erf* (*x*) is the complementary error

*Deff* (*hkl*) . The last relation from the mathematical

*FG*

⋅ *FT IC FC*

where

(36)

gg = g - g ò (34)


0 0 0 0

( ) 2( ) cos

*C CG C CG G G*

*ix x x x x x erfc*

ìï é ù æ ö g- - - g - - × Âí ç ê ú÷ - g g ïî ë û è ø

*ix x x x x x erfc*

é ù æ ö g+ - - g - - üï - Áê ú ç÷ ý g g ë û è ø ïþ

*C CG C CG G G*

0 0 0 0

( ) 2( ) sin

2

2

{ ( ( )) ( ( )) }

2

*L L is L*

*s i s s i s s*

2


= =

*erfc erfc*

point of view represents a Voigt distribution. If we take into account the properties of the Gauss

22 2 <sup>0</sup> <sup>0</sup> <sup>2</sup> 2 2 ( ) and ( ) *Ge G C C ix L L ix L L FL e G C FL e* - p -p g - p - pg = =

Explicit forms of experimental signal and true sample function normalized at *I*0*<sup>V</sup>* are given by

2 2

2

0 0 0

*G G*

é ù g- - - gg = <sup>×</sup> ê ú

*V C CG*

pg g ë û

( ) ( , , ) exp

*I xx x V x*

( ) 22 2 0 0 <sup>1</sup> <sup>2</sup> <sup>2</sup> (, , ) *C G G C ix x L L L V x FT e e G C*

*f s e e dL*

2 2 2 <sup>4</sup> 2 2 exp cos sin

¥


ò

( )

*π ∫* 0

*x e* <sup>−</sup>*<sup>t</sup>* <sup>2</sup>

and Cauchy distributions, the Voigt distribution can be generalized by relation

+¥


Based on relation (18), its Fourier transform is given by *FT V* = *FT IG*

0 0 ( , , ) ( ', , ) ( ', , ) ' *V x I x x I x x x dx GC G G GC C C*

é ù <sup>=</sup> <sup>Â</sup> - Á ê ú ë û

*<sup>λ</sup>* ), *erf* (*x*)= <sup>2</sup>

( )

<sup>2</sup>*<sup>π</sup>* <sup>2</sup> *<sup>ε</sup><sup>L</sup>* 2 *hkl h*0 2 *<sup>a</sup>* <sup>2</sup> , *<sup>γ</sup>* <sup>=</sup> <sup>1</sup>

The analytical expression of the Voigt distribution is

2 2

sin*θ*<sup>0</sup>

$$\begin{split} V\_{f}(\mathbf{x},\boldsymbol{\chi}\_{G,f},\boldsymbol{\chi}\_{C,f}) &= \frac{1}{\sqrt{\pi}\boldsymbol{\chi}\_{C,f}} \exp\left[\frac{\boldsymbol{\chi}\_{C,f}^{2} - (\mathbf{x} - \mathbf{x}\_{0:C} - \mathbf{x}\_{0G})^{2}}{\boldsymbol{\chi}\_{G,f}^{2}}\right]. \\ &\cdot \left\{ \Re\left[\mathrm{erfc}\left(\frac{\boldsymbol{\chi}\_{C,f} - i(\mathbf{x} - \mathbf{x}\_{0:C} - \mathbf{x}\_{0G})}{\boldsymbol{\chi}\_{G,f}}\right)\right] \cos\frac{2\boldsymbol{\chi}\_{C,f}\left(\mathbf{x} - \mathbf{x}\_{0:C} - \mathbf{x}\_{0G}\right)}{\boldsymbol{\chi}\_{G,f}^{2}} - \\ &- \Im\left[\mathrm{erfc}\left(\frac{\boldsymbol{\chi}\_{C,f} + i(\mathbf{x} - \mathbf{x}\_{0:C} - \mathbf{x}\_{0G})}{\boldsymbol{\chi}\_{G,f}}\right)\right] \sin\frac{2\boldsymbol{\chi}\_{C}\left(\mathbf{x} - \mathbf{x}\_{0:C} - \mathbf{x}\_{0G}\right)}{\boldsymbol{\chi}\_{G,f}^{2}}\right] \end{split} \tag{37}$$

Maximum value of true sample function is

$$V\_{\text{max}} = \frac{1}{\sqrt{\pi \gamma\_G}} \exp\left(\frac{\gamma\_c^2}{\gamma\_G^2}\right) \text{erfc}\left(\frac{\gamma\_c}{\gamma\_G}\right)$$

Voigt function is a probability density function and it displays the distribution of target values,

$$\begin{aligned} \int\_{-\infty}^{\infty} V(\mathbf{x}, \boldsymbol{\gamma}\_{G}, \boldsymbol{\gamma}\_{C}) d\mathbf{x} &= \frac{1}{\pi \sqrt{\pi}} \frac{\boldsymbol{\gamma}\_{C}}{\boldsymbol{\gamma}\_{G}} \int\_{-\infty}^{\infty} \left[ \int\_{-\infty}^{\infty} \frac{e^{-\left(\frac{\boldsymbol{\gamma}}{\boldsymbol{\gamma}\_{G}}\right)^{2}}}{\boldsymbol{\gamma}\_{C}^{2} + \left(\mathbf{x} - \mathbf{x}\right)^{2}} d\mathbf{x} d\mathbf{x} \right] d\mathbf{x} = \mathbf{1}, \\ \delta\_{V} &= \frac{\sqrt{\pi} \boldsymbol{\gamma}\_{G}}{\exp\left(\frac{\boldsymbol{\gamma}\_{C}}{\boldsymbol{\gamma}\_{G}}\right)^{2} \operatorname{erfc}\left(\frac{\boldsymbol{\gamma}\_{C}}{\boldsymbol{\gamma}\_{G}}\right)} \end{aligned} \tag{38}$$

and the convolution of two Voigt functions is also a Voigt function.

The integral width of a true sample function has the two components given by the Gauss and Cauchy contributions

$$\delta \mathcal{S}\_{G,f}^2 = \frac{1}{\pi} (\delta\_{G,h}^2 - \delta\_{G,\mathbf{g}}^2), \quad \delta\_{C,f} = \frac{1}{\pi} (\delta\_{C,h} - \delta\_{C,\mathbf{g}}) \tag{39}$$

Balzar and Popa are among the leading scientists in the field of Fourier analysis of X-ray diffraction profiles, and they suggested that each Gauss and Cauchy component contains information about the average crystallite size (*δ<sup>S</sup>* ) and distortion of the lattice (*δD*) as in [14]. From the algebraic point of view, they proposed the following relationship

$$
\delta\_{G,f}^2 = \delta\_{SG,f}^2 + \delta\_{DG,f}^2, \quad \delta\_{C,f} = \delta\_{SC,f} + \delta\_{DC,f} \tag{40}
$$

Based on the new concept introduced by them, the two components of the Fourier transform are given by the relations

$$F^S(L) = \mathbf{e}^{-\mathfrak{n}\mathbf{L}^2 \mathfrak{s}\_{\mathcal{H},f}^2 - 2\mathfrak{l}\mathfrak{l}\mathfrak{k}\_{\mathcal{K},f}}, \quad F^D(L) = \mathbf{e}^{-\mathfrak{n}\mathbf{L}^2 \mathfrak{s}\_{\mathcal{H},f}^2 - 2\mathfrak{l}\mathfrak{l}\mathfrak{k}\_{\mathcal{K},f}} \tag{41}$$

The particle size distribution function, *P(L)* is determined from the second derivative of straincorrected Fourier transform of the true sample function. The volume-weighted column-length *PV* and surface-weighted column-length *PS* distributions are given by the following [14]:

$$P\_V(L) = L \frac{d^2 F^S(L)}{dL^2} = 2L \left[ 2 \left( \pi L \delta\_{SG}^2 + \delta\_{SC} \right)^2 - \pi \delta\_{SG}^2 \right] F^S(L) \tag{42}$$

$$P\_S(L) = P\_V(L) = \frac{d^2 F^S(L)}{dL^2} = 2\left[2\left(\pi L \delta\_{SG}^2 + \delta\_{SC}\right)^2 - \pi \delta\_{SG}^2\right] F^S(L) \tag{43}$$

## **5. Experimental section, data analysis and results**

A series of four supported gold catalysts were studied by X-ray diffraction (XRD) in order to determine the average particle size of the gold, the microstrain of the lattice as well as the size and microstrain distribution functions by XRLP deconvolution using Fourier transform technique. The gold catalyst samples with up to 5 wt% gold content were prepared by impregnation of the SiO2 support with aqueous solution of HAuCl4×3H2O and homogeneous deposition-precipitation using urea as the precipitating agent method, respectively. The X-ray diffraction data of the supported gold catalysts displayed in Figure 3 were collected using a Rigaku horizontal powder diffractometer with rotated anode in Bragg-Brentano geometry with Ni-filtered Cu Kα radiation, λ = 1.54178 Å, at room temperature. The typical experimental conditions were: 60 sec for each step, initial angle 2*θ* = 32<sup>0</sup> , and a step of 0.020 , and each profile was measured at 2700 points. The XRD method is based on the deconvolution of the experi‐ mental XRLP (111) and (222) using Fourier transform procedure by fitting the XRLP with the Gauss, Cauchy, GFF and Voigt distributions. The Fourier analysis of XRLP validity depends strongly on the magnitude and nature of the errors propagated in the data analysis. The scientific literature treated three systematic errors: uncorrected constant background, trunca‐ tion, and effect of sampling for the observed profile at a finite number of points that appear in discrete Fourier analysis. In order to minimize propagation of these systematic errors, a global approximation of the XRLP is adopted instead of the discrete calculus. The reason for this choice was the simplicity and mathematical elegance of the analytical Fourier transform magnitude and the integral width of the true XRLP given by equations (20)-(24), (31), (34) and (38), as in [15]. The robustness of these approximations for the XRLP arises from the possibility of using the analytical forms of the Fourier transform instead of a numerical fast Fourier transform (FFT). It is well known that the validity of the numerical FFT depends drastically on the filtering technique what was adopted in [16]. In this way, the validity of the nanostruc‐ tural parameters is closely related to the accuracy of the Fourier transform magnitude of the true XRLP.

Based on the new concept introduced by them, the two components of the Fourier transform

The particle size distribution function, *P(L)* is determined from the second derivative of straincorrected Fourier transform of the true sample function. The volume-weighted column-length *PV* and surface-weighted column-length *PS* distributions are given by the following [14]:

( ) <sup>2</sup> <sup>2</sup> 2 2

( ) <sup>2</sup> <sup>2</sup> 2 2

A series of four supported gold catalysts were studied by X-ray diffraction (XRD) in order to determine the average particle size of the gold, the microstrain of the lattice as well as the size and microstrain distribution functions by XRLP deconvolution using Fourier transform technique. The gold catalyst samples with up to 5 wt% gold content were prepared by impregnation of the SiO2 support with aqueous solution of HAuCl4×3H2O and homogeneous deposition-precipitation using urea as the precipitating agent method, respectively. The X-ray diffraction data of the supported gold catalysts displayed in Figure 3 were collected using a Rigaku horizontal powder diffractometer with rotated anode in Bragg-Brentano geometry with Ni-filtered Cu Kα radiation, λ = 1.54178 Å, at room temperature. The typical experimental

was measured at 2700 points. The XRD method is based on the deconvolution of the experi‐ mental XRLP (111) and (222) using Fourier transform procedure by fitting the XRLP with the Gauss, Cauchy, GFF and Voigt distributions. The Fourier analysis of XRLP validity depends strongly on the magnitude and nature of the errors propagated in the data analysis. The scientific literature treated three systematic errors: uncorrected constant background, trunca‐ tion, and effect of sampling for the observed profile at a finite number of points that appear in discrete Fourier analysis. In order to minimize propagation of these systematic errors, a global approximation of the XRLP is adopted instead of the discrete calculus. The reason for this choice was the simplicity and mathematical elegance of the analytical Fourier transform magnitude and the integral width of the true XRLP given by equations (20)-(24), (31), (34) and (38), as in [15]. The robustness of these approximations for the XRLP arises from the possibility of using the analytical forms of the Fourier transform instead of a numerical fast Fourier transform (FFT). It is well known that the validity of the numerical FFT depends drastically

( ) ( ) 2 2 ( )

( ) () () 2 2 ( )

*V SG SC SG dF L PL L L L F L*

*S V SG SC SG dF L PL PL <sup>L</sup> F L*

2

2

*S*

*dL*

**5. Experimental section, data analysis and results**

conditions were: 60 sec for each step, initial angle 2*θ* = 32<sup>0</sup>

*S*

*dL*

2 2 2 2 , , , , 2 2 ( ) , () *S D L L SG f SC f L L DG f DC f FL e FL e* -p d - d -p d - d = = (41)

é ù = = p d + d - pd ê ú ë û (42)

é ù = = = p d + d - pd ê ú ë û (43)

*S*

*S*

, and a step of 0.020

, and each profile

are given by the relations

156 Fourier Transform - Signal Processing and Physical Sciences

Experimental relative intensities (111) with respect to 2*θ* values for (1) system are shown in Figure 4. The next steps consist in background correction of XRLP by polynomial procedures, finding the best parameters for the distributions adopted using the method of least squares or nonlinear fit, and then deconvoluting them using instrumental function. The main steps in the data analysis of the investigated systems are shown in Figure 4. Experimental relative intensities (111) with respect to 2*θ* values for (1) system are shown in Figure 4. The next steps consist in background correction of XRLP by polynomial procedures, finding the best parameters for the distributions adopted using the method of least squares or nonlinear fit, and then deconvoluting them using instrumental function. The main steps in the data analysis of the

Figure 4. Various stages of processing for X-ray line profile (111) of the sample (1) **Figure 4.** Various stages of processing for X-ray line profile (111) of the sample (1)

investigated systems are shown in Figure 4.

Experimental relative intensities (222) with respect to 2*θ* values for (6) system are shown in Figure 5. Experimental relative intensities (222) with respect to 2*θ* values for (6) system are shown in Figure 5.

The Fourier transforms normalized for the true sample function of the investigated samples (1) and (6) were calculated by three distinct methods, based on relations (28), (32) and (41), and are displayed in Figure 6.

The microstrain and particle size distribution functions determined by Fourier deconvolution of a single XRLP were calculated using equation (32), and are plotted in Figure 7.

The credibility of the parameters describing the investigated nanostructure systems depends primarily on the process of approximation of XRLP. This criterion is expressed by the root mean squares of residuals (*rmsr*) of data analysis and is given by relation

Figure 5. Various stages of processing for X-ray line profile (222) of the sample (6)

investigated systems are shown in Figure 4.

(6) were calculated by three distinct methods, based on relations (28), (32) and (41), and are displayed

∑ *i*=1 *<sup>N</sup>*points ( *yi*

Experimental relative intensities (111) with respect to 2*θ* values for (1) system are shown in Figure 4. The next steps consist in background correction of XRLP by polynomial procedures, finding the best parameters for the distributions adopted using the method of least squares or nonlinear fit, and then deconvoluting them using instrumental function. The main steps in the data analysis of the

> *calc* <sup>−</sup> *yi* exp)2

> > *σi* 2

*N*points− *N param*

*rmsr* =100

Figure 4. Various stages of processing for X-ray line profile (111) of the sample (1)

Figure 5. Various stages of processing for X-ray line profile (222) of the sample (6) The Fourier transforms normalized for the true samp **Figure 5.** Various stages of processing for X-ray line profile (222) of the sample (6) le function of the investigated samples (1) and

in Figure 6.

blue - general relation, red - GFF, green - Voigt distribution **Figure 6.** Fourier transform of true sample function of XRLP (111) and (222) for systems (1) and (6): blue - general rela‐ tion, red - GFF, green - Voigt distribution

Figure 6. Fourier transform of true sample function of XRLP (111) and (222) for systems (1) and (6):

The microstrain and particle size distribution functions determined by Fourier deconvolution of a

single XRLP were calculated using equation (32), and are plotted in Figure 7. The *rmsr* values for all distributions used in XRLP approximation process are given in Table 1. The *rmsr* values are closely related to the spectral noise of experimental data. Here it is shown that a model based on GFF and Voigt distribution may be more realistic and accurate.

The integral widths and FWHM of the true sample functions calculated for all distribu‐ tions were determined using the relations (21), (23), (31) and (38). Their values are presented in Table 2.

The credibility of the parameters describing the investigated nanostructure systems depends primarily on the process of approximation of XRLP. This criterion is expressed by the root mean squares of

Figure 7. Size and microstrain distribution functions of (111) XRLP for system (1)

residuals (*rmsr*) of data analysis and is given by relation

The microstrain and particle size distribution functions determined by Fourier deconvolution of a Experimental Data Deconvolution Based on Fourier Transform Applied in Nanomaterial Structure http://dx.doi.org/10.5772/59667 159

Figure 6. Fourier transform of true sample function of XRLP (111) and (222) for systems (1) and (6):

The Fourier transforms normalized for the true sample function of the investigated samples (1) and (6) were calculated by three distinct methods, based on relations (28), (32) and (41), and are displayed

in Figure 6.

**Figure 7.** Size and microstrain distribution functions of (111) XRLP for system (1)

Figure 7. Size and microstrain distribution functions of (111) XRLP for system (1)

blue - general relation, red - GFF, green - Voigt distribution

single XRLP were calculated using equation (32), and are plotted in Figure 7.


The credibility of the parameters describing the investigated nanostructure systems depends primarily on the process of approximation of XRLP. This criterion is expressed by the root mean squares of

**Table 1.** Values for *rmsr* for investigated samples

*rmsr* =100

Figure 4. Various stages of processing for X-ray line profile (111) of the sample (1)

∑ *i*=1 *<sup>N</sup>*points ( *yi*

Experimental relative intensities (222) with respect to 2*θ* values for (6) system are shown in Figure 5.

Figure 5. Various stages of processing for X-ray line profile (222) of the sample (6) The Fourier transforms normalized for the true samp **Figure 5.** Various stages of processing for X-ray line profile (222) of the sample (6) le function of the investigated samples (1) and (6) were calculated by three distinct methods, based on relations (28), (32) and (41), and are displayed

Figure 6. Fourier transform of true sample function of XRLP (111) and (222) for systems (1) and (6):

**Figure 6.** Fourier transform of true sample function of XRLP (111) and (222) for systems (1) and (6): blue - general rela‐

The *rmsr* values for all distributions used in XRLP approximation process are given in Table 1. The *rmsr* values are closely related to the spectral noise of experimental data. Here it is shown that a model based on GFF and Voigt distribution may be more realistic and accurate.

The integral widths and FWHM of the true sample functions calculated for all distribu‐ tions were determined using the relations (21), (23), (31) and (38). Their values are presented

The microstrain and particle size distribution functions determined by Fourier deconvolution of a

blue - general relation, red - GFF, green - Voigt distribution

tion, red - GFF, green - Voigt distribution

in Table 2.

single XRLP were calculated using equation (32), and are plotted in Figure 7.

Figure 7. Size and microstrain distribution functions of (111) XRLP for system (1)

residuals (*rmsr*) of data analysis and is given by relation

The credibility of the parameters describing the investigated nanostructure systems depends primarily on the process of approximation of XRLP. This criterion is expressed by the root mean squares of

Experimental relative intensities (111) with respect to 2*θ* values for (1) system are shown in Figure 4. The next steps consist in background correction of XRLP by polynomial procedures, finding the best parameters for the distributions adopted using the method of least squares or nonlinear fit, and then deconvoluting them using instrumental function. The main steps in the data analysis of the

investigated systems are shown in Figure 4.

158 Fourier Transform - Signal Processing and Physical Sciences

in Figure 6.

*calc* <sup>−</sup> *yi* exp)2

> *σi* 2

*N*points− *N param*

Because the experimental XRLP was measured for both (111) and (222), the surface-weighted column-length *PS* and volume-weighted column-length *PV* distribution functions were determined using relations (42,43) implemented in BREADTH software [17]. Additionally, it has found that the Gumbel distribution is the most adequate function for the global approxi‐ mation of both probabilities' curves, and the results are shown in Figure 8.

The global structural parameters obtained for the investigated samples are summarized in Table 3 and Table 4.


**Table 2.** Values for integral width and full width at half maximum for investigated samples

Figure 8. Surface-weighted column-length distribution function, *PS*, and volume-weighted columnlength distribution function, *PV*, for (5) and (6) systems **Figure 8.** Surface-weighted column-length distribution function, *PS*, and volume-weighted column-length distribution function, *PV*, for (5) and (6) systems


(5) 183 171 145 112 153 151 108 95 (6) 398 309 313 263 249 384 199 303 **Table 3.** Values for crystallite size determined by Scherrer method, and effective crystallite size using single XRLP approximations

Table 4. Values for the average crystallite size and microstrain using double Voigt approaches

*D <sup>V</sup>* ± Δ*DV*

(1) [111/222] 90±12 133±15. 0.211E-02 ± 0.152E-03 (3) [111/222] 69±13 104± 17 0.262E-02 ± 0.288E-03 (5) [111/222] 106±30 153±41 0.280E-02 ± 0.216E-03 (6) [111/222] 214±72 233±79 0.105E-02 ± 0.522E-03

Hydrogen chemisorption, transmission electron microscopy (TEM), magnetization, electronic paramagnetic resonance (EPR) and other methods could also be used to determine the average diameter of particles by taking into account a prior spherical form for the grains. By XRD method we can obtain the crystallite sizes that have different values for different crystallographic planes. There is a large difference between the particle size and the crystallite size due to the different physical meaning of the two concepts. It is possible that the particles of the supported gold catalysts are made

The size of the crystallites determined by equations (32) and (41), corresponding to (111) and (222) planes, have different values. The crystallite sizes *Sch <sup>D</sup>*<sup>111</sup> and *Sch <sup>D</sup>*222 are determined by the Scherrer method [18] without taking into account the microstrain of the lattice. The values *D111* and *D222* were determined by Fourier deconvolution method for single XRLP, while the averages of *DV* and *DS* were

[Å] ( ) ( ) 1/ <sup>2</sup> <sup>2</sup> 1/ <sup>2</sup> <sup>2</sup> / <sup>2</sup> ±Δ / <sup>2</sup> *<sup>V</sup> DV <sup>ε</sup> <sup>D</sup> <sup>ε</sup>*

Sample hkl *<sup>D</sup> <sup>S</sup>* <sup>±</sup> <sup>Δ</sup>*DS*

up of many gold crystallites.

[Å]


**Table 4.** Values for the average crystallite size and microstrain using double Voigt approaches

Hydrogen chemisorption, transmission electron microscopy (TEM), magnetization, electronic paramagnetic resonance (EPR) and other methods could also be used to determine the average diameter of particles by taking into account a prior spherical form for the grains. By XRD method we can obtain the crystallite sizes that have different values for different crystallo‐ graphic planes. There is a large difference between the particle size and the crystallite size due to the different physical meaning of the two concepts. It is possible that the particles of the supported gold catalysts are made up of many gold crystallites.

The size of the crystallites determined by equations (32) and (41), corresponding to (111) and (222) planes, have different values. The crystallite sizes *D*<sup>111</sup> *Sch* and *D*<sup>222</sup> *Sch* are determined by the Scherrer method [18] without taking into account the microstrain of the lattice. The values *D111* and *D222* were determined by Fourier deconvolution method for single XRLP, while the averages of *DV* and *DS* were calculated by a double Voigt approach. The difference between the crystallites' size can be explained by the fact that the analytical models are different due to the different approaches. This means that the geometry of the crystallites is not spherical [18]. The microstrain parameter of the lattice can also be correlated with the effective crystallite size in the following way: the value of the effective crystallite size increases when the micro‐ strain value decreases.

The main procedures of the SIZE.mws software dedicated to Fourier analysis of the XRLP by GFF and Voigt distributions written in Maple 11 language are presented in Appendix 2.

## **6. Conclusions**

**Sample hkl**

(1)

(3)

(5)

(6)

a\* represents FWHM

length distribution function, *PV*,

for (5) and (6) systems

*D*<sup>111</sup>

single XRLP approximations

up of many gold crystallites.

Table 4.

approximations

function, *PV*,

**2θ<sup>0</sup> [Deg]**

160 Fourier Transform - Signal Processing and Physical Sciences

δ [Deg]

a\* [Deg]

**Table 2.** Values for integral width and full width at half maximum for investigated samples

Figure 8. Surface-weighted column-length distribution function, *PS*,

**Figure 8.** Surface-weighted column-length distribution function, *PS*,

*Sch <sup>D</sup>*<sup>111</sup> D111

*Sch* D111 *<sup>D</sup>*<sup>222</sup>

[Å]

Sample hkl *<sup>D</sup> <sup>S</sup>* <sup>±</sup> <sup>Δ</sup>*DS*

for (5) and (6) systems

The global structural parameters obtained for the investigated samples are summarized in Table 3 and

**Sample GFF approximation Single Voigt approximation**

Table 3. Values for crystallite size determined by Scherrer method, and effective crystallite size using

(1) 180 168 180 139 145 154 118 145 (3) 143 135 139 130 116 118 98 139 (5) 183 171 145 112 153 151 108 95 (6) 398 309 313 263 249 384 199 303

*Sch* D222 *<sup>D</sup>*<sup>111</sup>

Sample GFF approximation Single Voigt approximation

(1) 180 168 180 139 145 154 118 145 (3) 143 135 139 130 116 118 98 139 (5) 183 171 145 112 153 151 108 95 (6) 398 309 313 263 249 384 199 303

*D <sup>V</sup>* ± Δ*DV*

(1) [111/222] 90±12 133±15. 0.211E-02 ± 0.152E-03 (3) [111/222] 69±13 104± 17 0.262E-02 ± 0.288E-03 (5) [111/222] 106±30 153±41 0.280E-02 ± 0.216E-03 (6) [111/222] 214±72 233±79 0.105E-02 ± 0.522E-03

Hydrogen chemisorption, transmission electron microscopy (TEM), magnetization, electronic paramagnetic resonance (EPR) and other methods could also be used to determine the average diameter of particles by taking into account a prior spherical form for the grains. By XRD method we can obtain the crystallite sizes that have different values for different crystallographic planes. There is a large difference between the particle size and the crystallite size due to the different physical meaning of the two concepts. It is possible that the particles of the supported gold catalysts are made

The size of the crystallites determined by equations (32) and (41), corresponding to (111) and (222) planes, have different values. The crystallite sizes *Sch <sup>D</sup>*<sup>111</sup> and *Sch <sup>D</sup>*222 are determined by the Scherrer method [18] without taking into account the microstrain of the lattice. The values *D111* and *D222* were determined by Fourier deconvolution method for single XRLP, while the averages of *DV* and *DS* were

*Sch <sup>D</sup>*<sup>111</sup> D111

[Å] ( ) ( ) 1/ <sup>2</sup> <sup>2</sup> 1/ <sup>2</sup> <sup>2</sup> / <sup>2</sup> ±Δ / <sup>2</sup> *<sup>V</sup> DV <sup>ε</sup> <sup>D</sup> <sup>ε</sup>*

*Sch <sup>D</sup>*<sup>222</sup> D222

**Table 3.** Values for crystallite size determined by Scherrer method, and effective crystallite size using single XRLP

Table 4. Values for the average crystallite size and microstrain using double Voigt approaches

δ [Deg]

**Distributions GFF Gauss Cauchy Voigt**

> δ [Deg]

a\* [Deg]

δ [Deg]

and volume-weighted column-

*Sch* D222

and volume-weighted column-length distribution

*Sch* D111 *<sup>D</sup>*<sup>222</sup>

*Sch <sup>D</sup>*<sup>222</sup> D222

a\* [Deg]

a\* [Deg]

111 38.291 0.644 0.540 0.638 0.599 0.738 0.469 0.713 0.537 222 81.780 0.862 0.715 0.854 0.803 1.025 0.653 0.927 0.765

111 38.220 0.802 0.672 0.798 0.750 0.919 0.585 0.891 0.668 222 81.763 1.028 0.860 1.011 0.950 1.181 0.752 1.095 0.912

111 38.211 0.631 0.529 0.621 0.583 0.723 0.460 0.701 0.519 222 81.801 1.052 0.873 1.067 1.003 1.225 0.780 1.185 0.890

111 38.292 0.363 0.304 0.363 0.341 0.426 0.271 0.374 0.335 222 81.872 0.506 0.423 0.498 0.468 0.613 0.390 0.528 0.461

> In the present chapter, it is shown that XRD analysis provides more information for under‐ standing the physical properties of nanomaterial structure. Powder X-ray diffraction is the cheapest and most reliable method compared with hydrogen chemisorptions, TEM techni‐ ques, magnetic measurements, EPR, etc. The main conclusions that can be drawn from these studies are:


## **Appendix 1**

Input data h.txt and g.txt files

```
`k`:=1;
line_h:= readline(`h.txt`):
line_g:= readline(`g.txt`):
while line <> 0 do
temp_h:= sscanf(line_h,`%8f%8f`):temp_g:=sscanf(line_g,`%8f %8f`):
printf(`%10.5f %10.5f`,temp_h[1],temp_h[2]): lprint():
printf(`%10.5f %10.5f`,temp_g[1],temp_g[2]): lprint():
twotheta_h[`k`]:=temp_h[1];
intensity_h[`k`]:=temp_h[2];
twotheta_g[`k`]:=temp_g[1];
intensity_g[`k`]:=temp_g[2];
line_h:= readline(`h.txt`):
line_g:= readline(`g.txt`):
`k`:=`k`+1;
end do;
`k`:=`k`-1;
p_h:=plot([twotheta_h[`ih`],intensity_h[`ih`],`ih`=1..k],col-
or=red,style=LINE,thickness=2,axes=boxed,gridlines,
labels=["2theta",""]):
p_g:=plot([twotheta_g[`ig`],intensity_g[`ig`],`ig`=1..k],col-
or=blue,style=LINE,thickness=2,axes=boxed,gridlines,
labels=["2theta",""]):
display({p_h,p_g});
deltatwotheta:=twotheta_h[2]-twotheta_h[1]:
```
h vector determination

```
twok:=2*k;
for `i` from 1 to k
do
h[`i`]:=intensity_h[`i`]:
end do:
```

```
h[`k`+1]:=0:
`j`:=1:
for `i` from `k`+2 to twok+1
do
h[`i`]:=intensity_h[`j`]:
`j`:=`j`+1:
end do:
print(h);
```
g array determination

**3.** Cauchy and Gauss distributions used for XRLP approximation give roughly structural

**4.** Powder X-ray diffraction gives the most detailed nanostructural results, such as: average crystallite size, microstrain, and distribution functions of crystallite size and microstrain;

**5.** Surface-weighted domain size depends only on Cauchy integral breadth, while volume-

**6.** To obtain valid structural results, it is important to have: a good S/N ratio of the experi‐ mental spectra, a good deconvolution technique for the experimental and instrumental

weighted domain size depends on Cauchy and Gauss integral breadths;

spectra, and an adequate computer package and programs for data analysis.

temp\_h:= sscanf(line\_h,`%8f%8f`):temp\_g:=sscanf(line\_g,`%8f %8f`):

p\_h:=plot([twotheta\_h[`ih`],intensity\_h[`ih`],`ih`=1..k],col-

p\_g:=plot([twotheta\_g[`ig`],intensity\_g[`ig`],`ig`=1..k],col-

or=red,style=LINE,thickness=2,axes=boxed,gridlines,

or=blue,style=LINE,thickness=2,axes=boxed,gridlines,

deltatwotheta:=twotheta\_h[2]-twotheta\_h[1]:

printf(`%10.5f %10.5f`,temp\_h[1],temp\_h[2]): lprint(): printf(`%10.5f %10.5f`,temp\_g[1],temp\_g[2]): lprint():

information;

162 Fourier Transform - Signal Processing and Physical Sciences

**Appendix 1**

Input data h.txt and g.txt files

`k`:=`k`+1; end do; `k`:=`k`-1;

h vector determination

do

twok:=2\*k;

end do:

line\_h:= readline(`h.txt`): line\_g:= readline(`g.txt`):

twotheta\_h[`k`]:=temp\_h[1]; intensity\_h[`k`]:=temp\_h[2]; twotheta\_g[`k`]:=temp\_g[1]; intensity\_g[`k`]:=temp\_g[2]; line\_h:= readline(`h.txt`): line\_g:= readline(`g.txt`):

labels=["2theta",""]):

labels=["2theta",""]): display({p\_h,p\_g});

for `i` from 1 to k

h[`i`]:=intensity\_h[`i`]:

while line <> 0 do

`k`:=1;

```
`j`:=1:
for `i` from -twok to -k
do
g[`i`]:=intensity_g[`j`]:
`j`:=`j`+1;
end do:
`j`:=1:
for `i` from -k to -1
do
g[`i`]:=intensity_g[`j`]:
`j`:=`j`+1:
end do:
g[0]:=0.:
for `i` from 1 to k
do
g[`i`]:=intensity_g[`i`]:
end do:
`j`:=1:
for `i` from k+1 to twok
do
g[`i`]:=intensity_g[`j`]:
`j`:=`j`+1:
end do:
print(g):
```
a matrix determination

```
for `i` from 1 to twok+1
do
for `j` from 1 to twok+1
do
a[`i`,`j`]:=0.:
end do:
end do:
`i1`:=0:
for `i` from -k to k
do
`j1`:=0:
`i1`:=`i1`+1:
for `j` from -k to k
do
```

```
`j1`:=`j1`+1:
a[`i1`,`j1`]:=g[`i`-`j`]*deltatwotheta;
end do:
end do:
print(a);
```
solving integral deconvolution equation by direct discretization

```
f:=linsolve(a,h):
for `i` from 1 to twok+1
do
twotheta_f[`i`]:=twotheta_h[1]+(`i`-1)*deltatwotheta:
intensity_f[`i`]:=eval(f[`i`]);
end do:
p_h:=plot([twotheta_h[`ihh`],intensity_h[`ihh`],`ihh`=1..k],col-
or=red,style=LINE,thickness=2,axes=boxed,gridlines,labels=["2theta",""]):
p_g:=plot([twotheta_g[`igg`],intensity_g[`igg`],`igg`=1..k],
color=blue,style=LINE,thickness=2,axes=boxed,gridlines,
labels=["2theta",""]):
p_f:=plot([twotheta_f[`iff`],intensity_f[`iff`],`iff`=1..twok+1],
color=green,style=LINE,thickness=2,axes=boxed,gridlines,
labels=["2theta",""]):
display({p_h,p_g,p_f});
fd:= fopen("f",WRITE,TEXT):
for `i` from 1 to k
do
fprintf(fd,"%g %g\n",twotheta_f[`i`],intensity_f[`i`]):
end do:
fclose(fd):
```
## **Appendix 2**

Fourier transform of true sample function procedure

```
f_GFF_freq:=proc(freq)
local arg_in,arg_sa;
arg_in:=(Pi*q_in)/(2*rho_in) + I *(Pi*Pi*freq)/rho_in;
arg_sa:=(Pi*q_sa)/(2*rho_sa) + I * (Pi*Pi*freq)/rho_sa;
(ampl_sa/ampl_in)*(rho_in/rho_sa)*cos(arg_in)/cos(arg_sa);
end:
```
Module of Fourier transform of true sample function procedure

```
FT_GFF_modul:=proc(freq)
local aux,bux,auxr_0,auxi_0;
aux:=evalc(Re(f_GFF_freq(freq))):
bux:=evalc(Im(f_GFF_freq(freq))):
aux:=aux*aux+bux*bux:
auxr_0:=evalc(Re(f_GFF_freq(0))):
auxi_0:=evalc(Im(f_GFF_freq(0))):
```

```
bux:=auxr_0*auxr_0+auxi_0*auxi_0:
sqrt(aux/bux):
end:
```
True sample function procedure

`j1`:=`j1`+1:

164 Fourier Transform - Signal Processing and Physical Sciences

f:=linsolve(a,h): for `i` from 1 to twok+1

labels=["2theta",""]):

labels=["2theta",""]): display({p\_h,p\_g,p\_f}); fd:= fopen("f",WRITE,TEXT):

for `i` from 1 to k

intensity\_f[`i`]:=eval(f[`i`]);

Fourier transform of true sample function procedure

f\_GFF\_freq:=proc(freq) local arg\_in,arg\_sa;

FT\_GFF\_modul:=proc(freq) local aux,bux,auxr\_0,auxi\_0; aux:=evalc(Re(f\_GFF\_freq(freq))): bux:=evalc(Im(f\_GFF\_freq(freq))):

aux:=aux\*aux+bux\*bux:

auxr\_0:=evalc(Re(f\_GFF\_freq(0))): auxi\_0:=evalc(Im(f\_GFF\_freq(0))):

end do: end do: print(a);

do

do

**Appendix 2**

end:

end do: fclose(fd):

end do:

a[`i1`,`j1`]:=g[`i`-`j`]\*deltatwotheta;

solving integral deconvolution equation by direct discretization

twotheta\_f[`i`]:=twotheta\_h[1]+(`i`-1)\*deltatwotheta:

p\_h:=plot([twotheta\_h[`ihh`],intensity\_h[`ihh`],`ihh`=1..k],col-

p\_f:=plot([twotheta\_f[`iff`],intensity\_f[`iff`],`iff`=1..twok+1],

p\_g:=plot([twotheta\_g[`igg`],intensity\_g[`igg`],`igg`=1..k], color=blue,style=LINE,thickness=2,axes=boxed,gridlines,

color=green,style=LINE,thickness=2,axes=boxed,gridlines,

fprintf(fd,"%g %g\n",twotheta\_f[`i`],intensity\_f[`i`]):

arg\_in:=(Pi\*q\_in)/(2\*rho\_in) + I \*(Pi\*Pi\*freq)/rho\_in; arg\_sa:=(Pi\*q\_sa)/(2\*rho\_sa) + I \* (Pi\*Pi\*freq)/rho\_sa; (ampl\_sa/ampl\_in)\*(rho\_in/rho\_sa)\*cos(arg\_in)/cos(arg\_sa);

Module of Fourier transform of true sample function procedure

or=red,style=LINE,thickness=2,axes=boxed,gridlines,labels=["2theta",""]):

```
f_GFF_s:=proc(s)
local arg1,arg2;
arg1:=(Pi/2)*(rho_sa/rho_in);
arg2:=(rho_sa*s);
(2/Pi)*(ampl_sa/ampl_in)*
rho_in*cos(arg1)*cosh(arg2)/(cosh(2*arg2)+cos(2*arg1));
end:
```
Integral width of true sample function procedure

```
int_width_GFF:=proc(rho_in,rho_sa)
local arg;
arg:=Pi*rho_sa/rho_in;
Pi/(2*rho_sa*cos(arg/2))*(cos(arg)+1);
end:
```
Moment of zero order for experimental X-ray line profile procedure

```
mu_0_GFF:=proc(ampl,rho,q)
local arg;
arg:=(Pi*q)/(2*rho);
(ampl/2)*(Pi/rho)*(1/cos(arg));
end:
```
Moment of first order for experimental X-ray line profile procedure

```
mu_1_GFF:=proc(rho,q)
local arg;
arg:=(Pi*q)/(2*rho);
(Pi/(2*rho))*tan(arg);
end:
```
Moment of second order for experimental X-ray line profile procedure

```
mu_2_GFF:=proc(rho,q)
local arg;
arg:=(Pi*q)/(2*rho);
((Pi/(2*rho))^2)*(1./(cos(arg)^2)+tan(arg)^2);
end:
```
Experimental X-ray line profile procedure approximated by GFF distribution

```
exp_profile_GFF:=proc(s)
local arg_q,arg_rho;
arg_q:=q*s; arg_rho:=rho*s;
(ampl/2)*(cosh(arg_q)+sinh(arg_q))/cosh(arg_rho);
end:
```
Instrumental X-ray line profile procedure determined by GFF distribution

```
inst_profile_GFF:=proc(s)
local arg_q,arg_rho;
arg_q:=q_in*s; arg_rho:=rho_in*s;
(ampl_in/2)*(cosh(arg_q)+sinh(arg_q))/cosh(arg_rho);
end:
```
Fourier transform procedure for general relation of true sample function developed by

Warren-Averbach theory

```
gen_function:=proc(freq)
exp(-beta_gen(fmin,fmax)*freq*freq-
gama_gen(fmin,fmax)*abs(freq));
end:
```
Procedure for experimental XRLP given by Voigt approximation

```
h_Voigt_function:=proc(s)
local arg1,arg2,arg3,arg4;
arg1:=(gama_h_c**2-s**2)/(gama_h_g**2);
arg2:=(gama_h_c-I*s)/gama_h_g;
arg3:=(gama_h_c+I*s)/gama_h_g;
arg4:=2.*gama_h_c*s/(gama_h_g**2);
amp_h/(sqrt(Pi)*gama_h_g)*exp(arg1)*
(Re(erfc(arg2))*cos(arg4)-Im(erfc(arg3))*sin(arg4));
end:
```
Procedure for instrumental XRLP given by Voigt approximation

```
g_Voigt_function:=proc(s)
local arg1,arg2,arg3,arg4;
arg1:=(gama_g_c**2-s**2)/(gama_g_g**2);
arg2:=(gama_g_c-I*s)/gama_g_g;
arg3:=(gama_g_c+I*s)/gama_g_g;
arg4:=2.*gama_g_c*s/(gama_g_g**2);
amp_g/(sqrt(Pi)*gama_g_g)*exp(arg1)*
(Re(erfc(arg2))*cos(arg4)-Im(erfc(arg3))*sin(arg4));
end:
```
Procedure for the true sample function calculated by Voigt approximation

```
f_Voigt_function:=proc(s)
local arg1,arg2,arg3,arg4;
arg1:=(gama_f_c**2-s**2)/(gama_f_g**2);
arg2:=(gama_f_c-I*s)/gama_f_g;
arg3:=(gama_f_c+I*s)/gama_f_g;
arg4:=2.*gama_f_c*s/(gama_f_g**2);
amp_h/amp_g/(sqrt(Pi)*gama_f_g)*exp(arg1)*(Re(erfc(arg2))*cos(arg4)-
Im(erfc(arg3))*sin(arg4));
end:
```
## **Acknowledgements**

Instrumental X-ray line profile procedure determined by GFF distribution

(ampl\_in/2)\*(cosh(arg\_q)+sinh(arg\_q))/cosh(arg\_rho);

Fourier transform procedure for general relation of true sample function developed by

inst\_profile\_GFF:=proc(s) local arg\_q,arg\_rho;

166 Fourier Transform - Signal Processing and Physical Sciences

gen\_function:=proc(freq)

h\_Voigt\_function:=proc(s) local arg1,arg2,arg3,arg4;

g\_Voigt\_function:=proc(s) local arg1,arg2,arg3,arg4;

f\_Voigt\_function:=proc(s) local arg1,arg2,arg3,arg4;

Im(erfc(arg3))\*sin(arg4));

arg2:=(gama\_f\_c-I\*s)/gama\_f\_g; arg3:=(gama\_f\_c+I\*s)/gama\_f\_g; arg4:=2.\*gama\_f\_c\*s/(gama\_f\_g\*\*2);

arg2:=(gama\_g\_c-I\*s)/gama\_g\_g; arg3:=(gama\_g\_c+I\*s)/gama\_g\_g; arg4:=2.\*gama\_g\_c\*s/(gama\_g\_g\*\*2); amp\_g/(sqrt(Pi)\*gama\_g\_g)\*exp(arg1)\*

arg2:=(gama\_h\_c-I\*s)/gama\_h\_g; arg3:=(gama\_h\_c+I\*s)/gama\_h\_g; arg4:=2.\*gama\_h\_c\*s/(gama\_h\_g\*\*2); amp\_h/(sqrt(Pi)\*gama\_h\_g)\*exp(arg1)\*

end:

Warren-Averbach theory

end:

end:

end:

end:

arg\_q:=q\_in\*s; arg\_rho:=rho\_in\*s;

exp(-beta\_gen(fmin,fmax)\*freq\*freqgama\_gen(fmin,fmax)\*abs(freq));

Procedure for experimental XRLP given by Voigt approximation

(Re(erfc(arg2))\*cos(arg4)-Im(erfc(arg3))\*sin(arg4));

(Re(erfc(arg2))\*cos(arg4)-Im(erfc(arg3))\*sin(arg4));

Procedure for the true sample function calculated by Voigt approximation

amp\_h/amp\_g/(sqrt(Pi)\*gama\_f\_g)\*exp(arg1)\*(Re(erfc(arg2))\*cos(arg4)-

Procedure for instrumental XRLP given by Voigt approximation

arg1:=(gama\_g\_c\*\*2-s\*\*2)/(gama\_g\_g\*\*2);

arg1:=(gama\_f\_c\*\*2-s\*\*2)/(gama\_f\_g\*\*2);

arg1:=(gama\_h\_c\*\*2-s\*\*2)/(gama\_h\_g\*\*2);

Financial support received from the European Union through the European Regional Devel‐ opment Fund, Project ID 1822/SMIS CSNR 48797 CETATEA, is gratefully acknowledged. In particular, one of the topics covered by the book *Fourier Transform of the Signals* will be a useful starting point in accomplishing one of its major objectives, energy recovery from ambient pollution. Additionally, the authors are grateful to the staff of Beijing Synchrotron Radiation Facilities for beam time and for their technical assistance in XRD measurements.

## **Author details**

Adrian Bot1 , Nicolae Aldea1\* and Florica Matei2

\*Address all correspondence to: naldea@itim-cj.ro

1 National Institute for Research and Development of Isotopic and Molecular Technologies, Cluj-Napoca, Romania

2 University of Agricultural Sciences and Veterinary Medicine, Cluj-Napoca, Romania

## **References**


## **Gabor-Fourier Analysis Gabor-Fourier Analysis**

Nafya Hameed Mohammad and Massoud Amini Nafya Hameed Mohammad1 and Massoud Amini2

Additional information is available at the end of the chapter Additional information is available at the end of the chapter

http://dx.doi.org/10.5772/60034 10.5772/60034

## **1. Introduction**

[7] Beniaminy I., Deutsch M. A Spline Based Method for Experimental Data Deconvolu‐

[8] Fredrikze H., Verkerk P. Comment on "A spline based method for experimental data

[9] Aldea N., Tiusan C., Zapotinschi R. A New Approach Used to Evaluation the Crys‐ tallite Size of Supported Metal Catalysts by Single X-Ray Profile Fourier Transform Implemented on Maple V. In: Borcherds P., Bubak M., Maksymowicz A., editors. 8th Joint EPS-APS International Conference on Physics Computing, PC '96 17-21 Septem‐

ber 1996, Krakow: Academic Computer Centre CYFRONET-KRAKOW; 1996.

[10] Balzar D., Ledbetter H. J. Voigt-Function Modeling in Fourier-Analysis of Size and Xray Diffraction Peaks-Peaks. Journal of Applied Crystallography 1993; 26(1) 97-103.

[11] Aldea N., Gluhoi A., Marginean P., Cosma C., Yaning X. Extended X-Ray Absorption Fine Structure and X-Ray Diffraction Studies on Supported Nickel Catalysts. Spectro‐

[12] Aldea N., Barz B., Pintea S., Matei F. Theoretical Approach Regarding Nanometrolo‐ gy of the Metal Nanoclusters Used in Heterogeneous Catalysis by Powder X-Ray Dif‐ fraction Method. Journal of Optoelectronics and Advanced Materials 2007; 9(10)

[13] Gradstein I. S., Rijik L. M. Tables of Integrals, Sums, Series, and Products. Moscow:

[14] Balzar D., Popa N. C. Crystallite Size and Residual Strain/Stress Modeling in Rietveld Refine. In: Meittemeijer E. J., Scardi P., editors. Diffraction Analysis of the Micro‐ structure of Materials. Berlin Heidelberg New York: Springer-Verlag; 2003, pp.

[15] Lazar M., Valer A., Pintea S., Barz B., Ducu C., Malinovschi V., Xie Yaning Aldea N. Preparation and Structural Characterization by XRD and XAS of the Supported Gold Catalysts. Journal of Optoelectronics and Advanced Materials 2008; 10(9) 2244-2251.

[16] Walker J. S. Fast Fourier Transform. 2nd ed. New York, London, Tokyo: Boca Raton

[17] Balzar D. *BREADTH-* a Program for Analyzing Diffraction Line Broadening. Journal

[18] Rednic V., Aldea N., Marginean P., Rada M., Bot A., Zhonghua W., Zhang J., Matei F. Heat Treatment Influence on the Structural Properties of Supported Ni Nanoclusters.

tion. Computer Physics Communications 1980; 21(2): 271-277.

chimica Acta Part 2000; 55(7) 997-1008.

168 Fourier Transform - Signal Processing and Physical Sciences

of Applied Crystallography 1995; 28(2) 244-245.

Metals and Materials International 2014; 20(4) 641-646.

3293-3296.

125-144.

CRC; 1997.

Fizmatgiz; 1962.

deconvolution". Computer Physics Communications 1981; 24(1), 5-7.

The notion of Gabor transform, named after Dennis Gabor [1], is a special case of the short-time Fourier transform. The Gabor analysis, as it stands now, is a rather new field, but the idea goes back quite some while. Dennis Gabor investigated in [1] the representation of a one dimensional signal in two dimensions, time and frequency. He suggested to represent a function by a linear combination of translated and modulated Gaussians. Interestingly, there is a tight connection between this approach and quantum mechanics (c.f. [2]). On the mathematical side, the representation of functions by other functions was further investigated, leading to the theory of atomic decompositions, developed by Feichtinger and Gröchenig [3].

Gabor transform and Gabor expansion have long been recognized as very useful tools for the signal processing, and it is because of this reason over the recent years, an increasing attention has been given to the study of them in engineering and applied Mathematics, see for instance [4, 5]. Borichev et al. [6] studied the stability problem for the Gabor expansions generated by a Gaussian function. In [7], Ascensi and Bruna proved that the unique Gabor atom with analytical Gabor space, the image of *L*2(**R**) under the Gabor transform, is the Gaussian function. The structure of Gabor and super Gabor spaces inside *L*2(**R**2*d*) is studied by Abreu [8]. Christensen [9] has done a comprehensive study of the Gabor system and has asked for the necessary and sufficient conditions to get a frame for *L*2(**R**).

Today Gabor analysis and the closely related wavelet analysis are considered topics in harmonic analysis. The basic idea behind wavelet analysis is that the notion of an orthonormal basis is not always useful. Sometimes it is more important for a decomposing set to have special properties, like good time frequency localization, than to have unique coefficients. This led to the concept of frames, which was introduced by Duffin and Schaefer in [10] and was made popular by Daubechies, and today is one of the most important foundations of Gabor theory and a fundamental subject in harmonic analysis.

©2012 Author(s), licensee InTech. This is an open access chapter distributed under 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.

Most examples of Gabor frames correspond to regular nets of points. That is, sets of the type {*e*2*<sup>π</sup>ibnth*(*<sup>t</sup>* <sup>−</sup> *am*)}*n*,*m*∈**Z***<sup>d</sup>* . One can usually find sufficient and necessary conditions for the existence of such kind of frames, with a variety of applications. For technical reasons, however, one needs to work with frames coming from irregular grids. One of the main purposes of this chapter is to study perturbations of irregular Gabor frames and the problem of stability.

On the other hand, the theory of nonharmonic Fourier series is concerned with the completeness and expansion properties of sets of complex exponentials {*eiλnt* } in *<sup>L</sup>p*[−*π*, *<sup>π</sup>*]. In 1952, Duffin and Schaeffer [10] used frames to study this theory, and later Young put together many results in his book [11]. Reid [12] proved that if {*λn*} is a sequence of real numbers whose differences are nondecreasing, then the set of complex exponentials {*eiλnt* } is a Riesz-Fischer sequence in *<sup>L</sup>*2[−*A*, *<sup>A</sup>*] for every *<sup>A</sup>* <sup>&</sup>gt; 0. Jaffard [13] investigated how the regularity of nonharmonic Fourier series is related to the spacing of their frequencies, and this is obtained by using a transformation which simultaneously captures the advantages of the Gabor and wavelet transforms.

In this chapter, we restate and prove some classical results of (nonharmonic) Fourier expansions for Gabor systems instead of sets of complex exponentials. Some of the results may be known or obtainable via Hilbert space methods, but the main advantage of this work is that it uses analytic methods and can be fully understood with elementary knowledge of functional and complex analysis in several variables [14, 15].

## **2. Preliminaries**

Let us introduce the notions and basic results, needed later in the chapter.

**Definition 2.1** We say that <sup>Λ</sup> <sup>=</sup> {*zj*}*j*∈**<sup>N</sup>** <sup>⊂</sup> **<sup>C</sup>***<sup>d</sup>* is a *separated set* if there exists *<sup>ε</sup>* <sup>&</sup>gt; 0 such that <sup>|</sup>*zi* <sup>−</sup> *zj*<sup>|</sup> <sup>≧</sup> *<sup>ε</sup>*, *<sup>i</sup>* �<sup>=</sup> *<sup>j</sup>*. The largest of such *<sup>ε</sup>* is called the *separation constant* of <sup>Λ</sup>. A finite union of separated sets is called a *relatively separated set.*

**Definition 2.2** A sequence of vectors {*xn*} in a normed space <sup>X</sup> is said to be *complete* if its linear span is dense in <sup>X</sup> , that is, if for each vector *<sup>x</sup>* and each *<sup>ε</sup>* <sup>&</sup>gt; 0 there is a finite linear combination *<sup>c</sup>*1*x*<sup>1</sup> <sup>+</sup> ··· <sup>+</sup> *cnxn* such that

$$\left\|\mathbf{x} - (\mathbf{c}\_1 \mathbf{x}\_1 + \dots + \mathbf{c}\_n \mathbf{x}\_n)\right\| < \varepsilon.$$

**Definition 2.3** A sequence { *fn*} in a Hilbert space *<sup>H</sup>* is said to be a *Bessel sequence* if

$$\sum\_{n=1}^{\infty} |\langle f, f\_n \rangle|^2 < \infty$$

for every element *<sup>f</sup>* <sup>∈</sup> *<sup>H</sup>*. It is called a *Riesz-Fischer sequence* if the moment problem

$$
\langle f\_{\nu} f\_n \rangle = c\_{n\nu} \quad n \ge 1
$$

admits at least one solution *<sup>f</sup>* <sup>∈</sup> *<sup>H</sup>* whenever {*cn*} ∈ *<sup>l</sup>* 2.

2 Fourier Transform

of stability.

the Gabor and wavelet transforms.

**2. Preliminaries**

Most examples of Gabor frames correspond to regular nets of points. That is, sets of the type {*e*2*<sup>π</sup>ibnth*(*<sup>t</sup>* <sup>−</sup> *am*)}*n*,*m*∈**Z***<sup>d</sup>* . One can usually find sufficient and necessary conditions for the existence of such kind of frames, with a variety of applications. For technical reasons, however, one needs to work with frames coming from irregular grids. One of the main purposes of this chapter is to study perturbations of irregular Gabor frames and the problem

On the other hand, the theory of nonharmonic Fourier series is concerned with the

In 1952, Duffin and Schaeffer [10] used frames to study this theory, and later Young put together many results in his book [11]. Reid [12] proved that if {*λn*} is a sequence of real numbers whose differences are nondecreasing, then the set of complex exponentials {*eiλnt*

is a Riesz-Fischer sequence in *<sup>L</sup>*2[−*A*, *<sup>A</sup>*] for every *<sup>A</sup>* <sup>&</sup>gt; 0. Jaffard [13] investigated how the regularity of nonharmonic Fourier series is related to the spacing of their frequencies, and this is obtained by using a transformation which simultaneously captures the advantages of

In this chapter, we restate and prove some classical results of (nonharmonic) Fourier expansions for Gabor systems instead of sets of complex exponentials. Some of the results may be known or obtainable via Hilbert space methods, but the main advantage of this work is that it uses analytic methods and can be fully understood with elementary knowledge of

**Definition 2.1** We say that <sup>Λ</sup> <sup>=</sup> {*zj*}*j*∈**<sup>N</sup>** <sup>⊂</sup> **<sup>C</sup>***<sup>d</sup>* is a *separated set* if there exists *<sup>ε</sup>* <sup>&</sup>gt; 0 such that <sup>|</sup>*zi* <sup>−</sup> *zj*<sup>|</sup> <sup>≧</sup> *<sup>ε</sup>*, *<sup>i</sup>* �<sup>=</sup> *<sup>j</sup>*. The largest of such *<sup>ε</sup>* is called the *separation constant* of <sup>Λ</sup>. A finite union

**Definition 2.2** A sequence of vectors {*xn*} in a normed space <sup>X</sup> is said to be *complete* if its linear span is dense in <sup>X</sup> , that is, if for each vector *<sup>x</sup>* and each *<sup>ε</sup>* <sup>&</sup>gt; 0 there is a finite linear

�*<sup>x</sup>* <sup>−</sup> (*c*1*x*<sup>1</sup> <sup>+</sup> ··· <sup>+</sup> *cnxn*)� <sup>&</sup>lt; *<sup>ε</sup>*.


�*<sup>f</sup>* , *fn*� <sup>=</sup> *cn*, *<sup>n</sup>* <sup>≧</sup> <sup>1</sup>

**Definition 2.3** A sequence { *fn*} in a Hilbert space *<sup>H</sup>* is said to be a *Bessel sequence* if

for every element *<sup>f</sup>* <sup>∈</sup> *<sup>H</sup>*. It is called a *Riesz-Fischer sequence* if the moment problem

∞ ∑ *n*=1 } in *<sup>L</sup>p*[−*π*, *<sup>π</sup>*].

}

completeness and expansion properties of sets of complex exponentials {*eiλnt*

functional and complex analysis in several variables [14, 15].

of separated sets is called a *relatively separated set.*

combination *<sup>c</sup>*1*x*<sup>1</sup> <sup>+</sup> ··· <sup>+</sup> *cnxn* such that

Let us introduce the notions and basic results, needed later in the chapter.

**Proposition 2.4** Let { *fn*} be a sequence in a Hilbert space *<sup>H</sup>*. Then

(i) { *fn*} is a Bessel sequence with bound *<sup>M</sup>* if and only if the inequality

$$\left\|\sum c\_n f\_n\right\|^2 \le M \sum |c\_n|^2$$

holds for every finite sequence of scalars {*cn*};

(ii) { *fn*} is a Riesz-Fischer sequence with bound *<sup>m</sup>* if and only if the inequality

$$m\sum |c\_n|^2 \le \left\| \sum c\_n f\_n \right\|^2$$

holds for every finite sequence of scalars {*cn*}.

**Remark 2.5** For a sequence { *fn*} in a Hilbert space *<sup>H</sup>*, the moment problem

$$\langle f\_{\nu} f\_{n} \rangle = c\_{n\nu} \quad n \ge 1$$

has at most one solution for every choice of the scalars {*cn*} if and only if { *fn*} is complete.

**Definition 2.6** A countable family { *fk*}*k*∈*<sup>I</sup>* in a separable Hilbert space *<sup>H</sup>* is a *frame* for *<sup>H</sup>* if there exist constants *A* and *B* such that 0 < *A* ≦ *B* < ∞ and

$$A\|f\|^2 \le \sum\_{k \in I} |\langle f\_{\prime}f\_k \rangle|^2 \le B\|f\|^2, \quad f \in H.$$

*A*, *B* are called the *lower* and *upper frame bounds* respectively. They are not unique: the biggest lower bound and the smallest upper bound are called the *optimal frame bounds*. Every element in *H* has at least one representation as an infinite linear combination of the frame elements.

**Definition 2.7** Let *<sup>c</sup>* <sup>∈</sup> **<sup>R</sup>***d*, the unitary operators *Tc* and *Mc* on *<sup>L</sup>*2(**R***d*) defined by *Tc <sup>f</sup>*(*t*) = *<sup>f</sup>*(*<sup>t</sup>* <sup>−</sup> *<sup>c</sup>*) and *Mc <sup>f</sup>*(*t*) = *<sup>e</sup>*2*πict <sup>f</sup>*(*t*) are called the *Translation* and *Modulation* operator, respectively. For a discrete set <sup>Λ</sup> <sup>=</sup> {*zj*}*j*∈**<sup>Z</sup>** in **<sup>C</sup>***<sup>d</sup>* and a fixed nonzero window function *<sup>h</sup>* <sup>∈</sup> *<sup>L</sup>*2(**R***d*), we define the *Gabor system G*(*h*, <sup>Λ</sup>) as:

$$G(h, \Lambda) = \{M\_{\mathcal{Y}} T\_{\mathcal{X}} h(t) = e^{2\pi iyt} h(t - \mathbf{x}) ; \; \mathbf{x} + i\mathbf{y} \in \Lambda\}.$$

For simplicity we denote *<sup>e</sup>*2*<sup>π</sup>iyth*(*<sup>t</sup>* <sup>−</sup> *<sup>x</sup>*) by *hz*(*t*), where *<sup>z</sup>* <sup>=</sup> *<sup>x</sup>* <sup>+</sup> *iy*. Gabor systems were first introduced by Gabor [1] in 1946 for signal processing, and is still widely used. A Gabor system is said to be *exact* in *L*2(**R***d*) if it is complete, but fails to be complete on the removal of any one term.

If *G*(*h*, Λ) is a frame for *L*2(**R***d*), it is called a *Gabor frame* or *Weyl-Heisenberg frame*.

**Definition 2.8** Let *f* be an entire function. For *r* > 0, the *maximum modulus function* is *<sup>M</sup>*(*r*) = max{| *<sup>f</sup>*(*z*)<sup>|</sup> : <sup>|</sup>*z*<sup>|</sup> <sup>=</sup> *<sup>r</sup>*}. Unless *<sup>f</sup>* is a constant of modulus less than or equal to 1, its *order*, which is denoted by *ρ*, is defined by

$$\rho = \limsup\_{r \to \infty} \frac{\log \log M(r)}{\log r}.$$

Simple examples of functions of finite order include *ez*, sin *z*, and cos *z*, all of which are of order 1, and cos √*z*, which is of order <sup>1</sup> <sup>2</sup> . Every polynomial is of order 0; the order of a constant function is of course 0 and the function *ee<sup>z</sup>* is of infinite order.

**Remark 2.9** An entire function has an order of growth <sup>≦</sup> *<sup>ρ</sup>* if <sup>|</sup> *<sup>f</sup>*(*z*)<sup>|</sup> <sup>≦</sup> *A eB*|*z*<sup>|</sup> *ρ* .

The following is the fundamental factorization theorem for entire functions of finite order. It is due to Hadamard who used the result in his celebrated proof of the Prime Number Theorem. It is one of the classical theorems in function theory.

**Theorem 2.10 (Hadamard Factorization Theorem)** Let *f* be an entire function of finite order *<sup>ρ</sup>*, {*zn*} be the zeros of *<sup>f</sup>* different from 0, *<sup>k</sup>* be the order of zero of *<sup>f</sup>* at the origin, and

$$f(z) = z^k e^{\mathbf{g}(z)} \prod\_{n=1}^{\infty} (1 - \frac{z}{z\_n})$$

be its canonical Factorization, then *g*(*z*) is a polynomial of degree no longer than *ρ*.

**Definition 2.11** The *(Bargmann-)Fock space*, <sup>F</sup>(**C***d*), is the Hilbert space of all entire functions *f* on **C***<sup>d</sup>* for which

$$\|f\|\_{\mathcal{F}}^2 = \int\_{\mathbf{C}^d} |f(z)|^2 e^{-\pi |z|^2} dz$$

is finite. The natural inner product on <sup>F</sup>(**C***d*) is defined by

$$\langle f, g \rangle \,\boldsymbol{\mathcal{F}} = \int\_{\mathbf{C}^d} f(z) \overline{g(z)} e^{-\pi |z|^2} \, dz; \quad f, g \in \mathcal{F}(\mathbf{C}^d).$$

The *Bargmann transform* of a function *<sup>f</sup>* <sup>∈</sup> *<sup>L</sup>*2(**R***d*) is the function *B f* on **<sup>C</sup>***<sup>d</sup>* defined by

$$Bf(z) = 2^{\frac{d}{4}}e^{-\frac{\pi}{2}z^2} \int\_{\mathbb{R}^d} f(t)e^{-\pi t^2}e^{2\pi tz} \,dt.$$

**Definition 2.12** Fix a function *<sup>h</sup>* <sup>∈</sup> *<sup>L</sup>*2(**R***d*) (called the window function). The *Gabor transform* with respect to the window *h* is the isomorphic inclusion

$$V\_{\hbar}: L^2(\mathbb{R}^d) \longrightarrow L^2(\mathbb{C}^d)\_{\prime}$$

defined by

4 Fourier Transform

of any one term.

*f* on **C***<sup>d</sup>* for which

*order*, which is denoted by *ρ*, is defined by

order 1, and cos √*z*, which is of order <sup>1</sup>

constant function is of course 0 and the function *ee<sup>z</sup>*

system is said to be *exact* in *L*2(**R***d*) if it is complete, but fails to be complete on the removal

**Definition 2.8** Let *f* be an entire function. For *r* > 0, the *maximum modulus function* is *<sup>M</sup>*(*r*) = max{| *<sup>f</sup>*(*z*)<sup>|</sup> : <sup>|</sup>*z*<sup>|</sup> <sup>=</sup> *<sup>r</sup>*}. Unless *<sup>f</sup>* is a constant of modulus less than or equal to 1, its

Simple examples of functions of finite order include *ez*, sin *z*, and cos *z*, all of which are of

The following is the fundamental factorization theorem for entire functions of finite order. It is due to Hadamard who used the result in his celebrated proof of the Prime Number

**Theorem 2.10 (Hadamard Factorization Theorem)** Let *f* be an entire function of finite order

**Definition 2.11** The *(Bargmann-)Fock space*, <sup>F</sup>(**C***d*), is the Hilbert space of all entire functions

∏*n*=1

(<sup>1</sup> <sup>−</sup> *<sup>z</sup> zn* )

*<sup>ρ</sup>*, {*zn*} be the zeros of *<sup>f</sup>* different from 0, *<sup>k</sup>* be the order of zero of *<sup>f</sup>* at the origin, and

*<sup>f</sup>*(*z*) = *<sup>z</sup>keg*(*z*) <sup>∞</sup>

be its canonical Factorization, then *g*(*z*) is a polynomial of degree no longer than *ρ*.

*f*(*z*)*g*(*z*)*e*

The *Bargmann transform* of a function *<sup>f</sup>* <sup>∈</sup> *<sup>L</sup>*2(**R***d*) is the function *B f* on **<sup>C</sup>***<sup>d</sup>* defined by

<sup>−</sup>*π*|*z*<sup>|</sup> 2

*dz*; *<sup>f</sup>* , *<sup>g</sup>* ∈ F(**C***d*).

� *<sup>f</sup>* �<sup>2</sup> F = **C***d* <sup>|</sup> *<sup>f</sup>*(*z*)<sup>|</sup> 2*e* <sup>−</sup>*π*|*z*<sup>|</sup> 2 *dz*,

 **C***d*

is finite. The natural inner product on <sup>F</sup>(**C***d*) is defined by

�*<sup>f</sup>* , *<sup>g</sup>*�F <sup>=</sup>

log log *M*(*r*) log *<sup>r</sup>* .

<sup>2</sup> . Every polynomial is of order 0; the order of a

*ρ* .

is of infinite order.

If *G*(*h*, Λ) is a frame for *L*2(**R***d*), it is called a *Gabor frame* or *Weyl-Heisenberg frame*.

*ρ* = lim sup *<sup>r</sup>*→<sup>∞</sup>

**Remark 2.9** An entire function has an order of growth <sup>≦</sup> *<sup>ρ</sup>* if <sup>|</sup> *<sup>f</sup>*(*z*)<sup>|</sup> <sup>≦</sup> *A eB*|*z*<sup>|</sup>

Theorem. It is one of the classical theorems in function theory.

$$V\_{\hbar}f(z) = \langle f(t), h\_{z}(t) \rangle = 2^{d/4} \int\_{\mathbb{R}^d} f(t) \overline{h(t-x)} e^{-2\pi i t y} \, dt$$

for every *<sup>f</sup>* <sup>∈</sup> *<sup>L</sup>*2(**R***d*) and *<sup>z</sup>* <sup>=</sup> *<sup>x</sup>* <sup>+</sup> *iy* <sup>∈</sup> **<sup>C</sup>***d*. The following subspace of *<sup>L</sup>*2(**C***d*) which is the image of *L*2(**R***d*) under the Gabor transform with the window *h*,

$$\mathcal{G}\_h = \{ V\_h f : f \in L^2(\mathbb{R}^d) \}\_{\prime \prime}$$

is called *Gabor space* or *model space*. A simple calculation shows that the Bargmann transform is related to the Gabor transform with the Gaussian window *<sup>g</sup>*(*t*) = <sup>2</sup>*d*/4*e*−*π<sup>t</sup>* 2 in *L*2(**R***d*) by the formula

$$V\_{\mathcal{S}}f(\mathbf{x} - i\mathbf{y}) = e^{i\pi \mathbf{x} \mathbf{y}} e^{-\pi \frac{|\mathbf{x} + i\mathbf{y}|^2}{2}} (\mathcal{B}f)(\mathbf{x} + i\mathbf{y}). \tag{1}$$

For more details we refer the reader to [2, 7, 8, 17].

## **3. Gabor Expansion**

Here we discuss the fundamental completeness properties of the Gabor systems. The most extensive results in the case of the sets of complex exponentials {*eiλnt* } over a finite interval of the real axis were obtained by Paley and Wiener [16]. At the same time, we will be laying the groundwork for a more penetrating investigation of nonharmonic Gabor expansions in *L*2(**R**2).

Let {(*λn*, *<sup>µ</sup>n*)}*n*∈**<sup>Z</sup>** be an arbitrary countable subset of **<sup>R</sup>**<sup>2</sup> and

$$\left\{\phi\_{\boldsymbol{n}}(\boldsymbol{\xi})\right\}\_{\boldsymbol{n}\in\mathbb{Z}} = \left\{M\_{\mu\_{\boldsymbol{n}}}T\_{\lambda\_{\boldsymbol{n}}}g(\boldsymbol{\xi})\right\}\_{\boldsymbol{n}\in\mathbb{Z}} = \left\{\sqrt{2}\,e^{2\pi i\mu\_{\boldsymbol{n}}\boldsymbol{\xi} - \pi(\boldsymbol{\xi}-\lambda\_{\boldsymbol{n}})^{2}}\right\}\_{\boldsymbol{n}\in\mathbb{Z}}\tag{2}$$

where *<sup>ξ</sup>* <sup>∈</sup> **<sup>R</sup>**<sup>2</sup> or **<sup>C</sup>**, be the corresponding Gabor system with respect to the Gaussian window *<sup>g</sup>* in *<sup>L</sup>*2(**R**2). If {*ϕn*}*n*∈**<sup>Z</sup>** is incomplete in *<sup>L</sup>*2(**R**2) then the closed linear span <sup>M</sup> of {*ϕn*}*n*∈**<sup>Z</sup>** is a proper subspace of *L*2(**R**2). By Hahn-Banach Theorem there exists a function *F* in *L*2(**R**2)

such that *<sup>F</sup>*|M <sup>=</sup> 0 and *<sup>F</sup>* �<sup>=</sup> 0. Riesz Representation Theorem implies that *<sup>F</sup>* <sup>=</sup> *<sup>F</sup><sup>ϕ</sup>* for some *ϕ* in *L*2(**R**2) and

$$F(h) = F\_{\mathfrak{P}}(h) = \int\_{\mathbb{R}^2} h \, \mathfrak{p} \, d\mathfrak{F}; h \in L^2(\mathbb{R}^2).$$

For (*z*, *<sup>w</sup>*) <sup>∈</sup> **<sup>C</sup>**<sup>2</sup> take

$$f(z, w) = \sqrt{2} \int\_{\mathbb{R}^2} e^{2\pi i w \xi - \pi(\xi - z)^2} \, \varphi(\xi) \, d\xi,\tag{3}$$

then *<sup>f</sup>*(*λn*, *<sup>µ</sup>n*) = *<sup>F</sup>*(*ϕn*) = 0 (since *<sup>F</sup>*|M <sup>=</sup> 0).

**Remark 3.1** The system (2) is incomplete in *L*2(**R**2) if and only if there exists a nontrivial entire function of the form (3) in the Gabor space <sup>G</sup>*g*, which is zero for every (*λn*, *<sup>µ</sup>n*). Furthermore, since

$$f(z, w) = V\_{\mathcal{S}} \varphi(z, -w) = e^{i\pi zw} e^{-\pi \frac{|z|^2 + |w|^2}{2}} (B\varphi)(z, w)\_{\sigma}$$

we have

$$|f(z, w)| \le ||\varrho||\_2 \, e^{\frac{\varpi}{2}|(z, w)|^2}.$$

**Theorem 3.2** Let {*λn*}*n*∈**<sup>Z</sup>** be a symmetric sequence of real numbers (*λ*−*<sup>n</sup>* <sup>=</sup> <sup>−</sup>*λn*). If the Gabor system

$$\left\{ \sqrt[4]{\mathbb{Z}} \, e^{2\pi i \lambda\_{\pi} t - \pi (t - \lambda\_{\pi})^2} \right\}\_{n \in \mathbb{Z}} \tag{4}$$

is exact in *L*2(**R**), then the product

$$\prod\_{n=1}^{\infty} \left( 1 - \frac{z^2}{\lambda\_n^2} \right) e^{\frac{z^2}{\lambda\_n^2}}$$

converges to an entire function which belongs to the Gabor space with Gaussian window in *L*2(**R**).

**Proof.** By Remark 3.1, if the system (4) is exact, then there exists an entire function *f*(*z*) in the Gabor space <sup>G</sup>*<sup>g</sup>* such that *<sup>f</sup>*(*λn*) = 0 for *<sup>n</sup>* �<sup>=</sup> 0, and

$$f(z) = \sqrt[4]{2} \int\_{\mathbb{R}} e^{2\pi izt - \pi(t-z)^2} \,\varphi(t) \,dt; \quad \varphi \in L^2(\mathbb{R}).$$

Since *<sup>f</sup>*(*λn*) = 0 for *<sup>n</sup>* �<sup>=</sup> 0 and the sequence {*λn*} is symmetric, *<sup>ϕ</sup>*(−*t*) has the same orthogonality properties as *ϕ*(*t*). But by Remark 2.5, *ϕ*(*t*) is unique, so *ϕ*(*t*) must be even.

10.5772/60034

Hence *<sup>f</sup>*(*z*) is even. Now *<sup>f</sup>*(*z*) vanishes only at the *<sup>λ</sup><sup>n</sup>* with *<sup>n</sup>* �<sup>=</sup> 0. Indeed, if *<sup>f</sup>*(*z*) vanishes at *z* = *γ*, then the function

6 Fourier Transform

*ϕ* in *L*2(**R**2) and

For (*z*, *<sup>w</sup>*) <sup>∈</sup> **<sup>C</sup>**<sup>2</sup> take

Furthermore, since

we have

Gabor system

*L*2(**R**).

is exact in *L*2(**R**), then the product

the Gabor space <sup>G</sup>*<sup>g</sup>* such that *<sup>f</sup>*(*λn*) = 0 for *<sup>n</sup>* �<sup>=</sup> 0, and

*f*(*z*) = √<sup>4</sup> 2

 **R** *e*

such that *<sup>F</sup>*|M <sup>=</sup> 0 and *<sup>F</sup>* �<sup>=</sup> 0. Riesz Representation Theorem implies that *<sup>F</sup>* <sup>=</sup> *<sup>F</sup><sup>ϕ</sup>* for some

**Remark 3.1** The system (2) is incomplete in *L*2(**R**2) if and only if there exists a nontrivial entire function of the form (3) in the Gabor space <sup>G</sup>*g*, which is zero for every (*λn*, *<sup>µ</sup>n*).

<sup>|</sup> *<sup>f</sup>*(*z*, *<sup>w</sup>*)<sup>|</sup> <sup>≦</sup> �*ϕ*�<sup>2</sup> *<sup>e</sup>*

**Theorem 3.2** Let {*λn*}*n*∈**<sup>Z</sup>** be a symmetric sequence of real numbers (*λ*−*<sup>n</sup>* <sup>=</sup> <sup>−</sup>*λn*). If the

<sup>2</sup>*πiλnt*−*π*(*t*−*λn*)<sup>2</sup>

converges to an entire function which belongs to the Gabor space with Gaussian window in

**Proof.** By Remark 3.1, if the system (4) is exact, then there exists an entire function *f*(*z*) in

Since *<sup>f</sup>*(*λn*) = 0 for *<sup>n</sup>* �<sup>=</sup> 0 and the sequence {*λn*} is symmetric, *<sup>ϕ</sup>*(−*t*) has the same orthogonality properties as *ϕ*(*t*). But by Remark 2.5, *ϕ*(*t*) is unique, so *ϕ*(*t*) must be even.

<sup>2</sup>*πizt*−*π*(*t*−*z*)<sup>2</sup>

*<sup>i</sup><sup>π</sup>zwe* <sup>−</sup>*<sup>π</sup>* <sup>|</sup>*z*<sup>|</sup> <sup>2</sup>+|*w*<sup>|</sup> 2

> *π* <sup>2</sup> <sup>|</sup>(*z*,*w*)<sup>|</sup> 2 .

*<sup>ϕ</sup>*(*t*) *dt*; *<sup>ϕ</sup>* <sup>∈</sup> *<sup>L</sup>*2(**R**).

 *e z*2 *λ*2 *n*

*<sup>h</sup> <sup>ϕ</sup> <sup>d</sup>ξ*; *<sup>h</sup>* <sup>∈</sup> *<sup>L</sup>*2(**R**2).

<sup>2</sup> (*Bϕ*)(*z*, *w*),

*ϕ*(*ξ*) *dξ*, (3)

*<sup>n</sup>*∈**<sup>Z</sup>** (4)

<sup>2</sup>*πiwξ*−*π*(*ξ*−*z*)<sup>2</sup>

 **R**2

*F*(*h*) = *Fϕ*(*h*) =

*f*(*z*, *w*) = √

*<sup>f</sup>*(*z*, *<sup>w</sup>*) = *Vgϕ*(*z*, <sup>−</sup>*w*) = *<sup>e</sup>*

<sup>√</sup><sup>4</sup> <sup>2</sup> *<sup>e</sup>*

∞ ∏*n*=1  <sup>1</sup> <sup>−</sup> *<sup>z</sup>*<sup>2</sup> *λ*2 *n*

then *<sup>f</sup>*(*λn*, *<sup>µ</sup>n*) = *<sup>F</sup>*(*ϕn*) = 0 (since *<sup>F</sup>*|M <sup>=</sup> 0).

2 **R**2 *e*

$$\tilde{f}(z) = \frac{zf(z)}{z - \gamma}$$

would also belong to <sup>G</sup>*<sup>g</sup>* and would vanish at every *<sup>λ</sup>n*. The system (4) would then be incomplete in *L*2(**R**), contrary to hypothesis.

Let us observe that the function ˜ *<sup>f</sup>* belongs to <sup>G</sup>*g*. Since the Bargmann transform is related to the Gabor transform by the formula (1), it is sufficient to show that the function *ei<sup>π</sup>xye π* <sup>2</sup> (|*x*<sup>|</sup> <sup>2</sup>+|*y*<sup>|</sup> <sup>2</sup>) ˜ *<sup>f</sup>*(*z*); *<sup>z</sup>* <sup>=</sup> *<sup>x</sup>* <sup>+</sup> *iy*, belongs to the Fock space <sup>F</sup>(**C**). In other words, we must show that the integral

$$\int\_{\mathbb{C}} \frac{|z|^2}{|z-\gamma|^2} \left| f(z) e^{i\pi \text{xy}} e^{\frac{\pi}{2}(|\mathbf{x}|^2+|y|^2)} \right|^2 e^{-\pi(|\mathbf{x}|^2+|y|^2)} d\mathbf{x} \, dy; \quad z=\mathbf{x}+i\mathbf{y}, \bar{z}$$

is finite. Since lim*z*→<sup>∞</sup> <sup>|</sup> *<sup>z</sup> <sup>z</sup>* <sup>−</sup> *<sup>γ</sup>* <sup>|</sup> <sup>=</sup> 1, we have <sup>|</sup> *<sup>z</sup> <sup>z</sup>* <sup>−</sup> *<sup>γ</sup>* <sup>|</sup> <sup>≦</sup> 3/2 outside a square *<sup>T</sup>* with complement *Tc*. Thus the above integral is no larger than

$$\begin{split} &\int\_{T} \frac{|z|^2}{|z-\gamma|^2} \Big| f(z) e^{i\pi xy} e^{\frac{\pi}{2} (|x|^2+|y|^2)} \Big|^2 e^{-\pi (|x|^2+|y|^2)} dx \, dy \\ &+ 9/4 \int\_{T^c} |f(z) e^{i\pi xy} e^{\frac{\pi}{2} (|x|^2+|y|^2)} \Big|^2 e^{-\pi (|x|^2+|y|^2)} dx \, dy \\ &\leq \int\_{T} \frac{|z|^2}{|z-\gamma|^2} \Big| f(z) e^{i\pi xy} e^{\frac{\pi}{2} (|x|^2+|y|^2)} \Big|^2 e^{-\pi (|x|^2+|y|^2)} dx \, dy \\ &+ 9/4 \int\_{\mathbb{C}} \Big| f(z) e^{i\pi xy} e^{\frac{\pi}{2} (|x|^2+|y|^2)} \Big|^2 e^{-\pi (|x|^2+|y|^2)} dx \, dy. \end{split}$$

In the last expression, since *T* is compact the first integral is finite, and since the function *f*(*z*)*ei<sup>π</sup>xye π* <sup>2</sup> (|*x*<sup>|</sup> <sup>2</sup>+|*y*<sup>|</sup> <sup>2</sup>) is in the Fock space <sup>F</sup>(**C**), so is the second integral. Next since

$$|f(z)| \le \|\varphi\|\_2 \, e^{\frac{\pi}{2}|z|^2} \, \rho$$

the order of growth of *f* is at most 2, and by Hadamard's factorization theorem,

$$f(z) = e^{Az} \prod\_{n=1}^{\infty} \left( 1 - \frac{z^2}{\lambda\_n^2} \right) e^{\frac{z^2}{\lambda\_n^2}}; \quad A \in \mathbb{R}.$$

Since *f*(*z*) and the canonical product are both even, *A* = 0 and

$$f(z) = \prod\_{n=1}^{\infty} \left( 1 - \frac{z^2}{\lambda\_n^2} \right) e^{\frac{z^2}{\lambda\_n^2}}.$$

We have the following version of Plancherel-Pólya theorem. We give the proof which is similar to [11, Th. 2.16] for the sake of completeness.

**Theorem 3.3 (Plancherel-Pólya).** If *f*(*z*) is an entire function of order of growth≦ *τ* and if for some positive number *p*,

$$\int\_{-\infty}^{\infty} |f(x)|^p \, dx < \infty,$$

then

$$\int\_{-\infty}^{\infty} |f(\mathbf{x} + i\mathbf{y})|^p \, d\mathbf{x} \le e^{p\tau|y|} \int\_{-\infty}^{\infty} |f(\mathbf{x})|^p \, d\mathbf{x}.$$

The proof will require two preliminary lemmas. Suppose that *q*(*z*) is a non constant continuous function in the closed upper half-plane, Im*z* ≧ 0, and analytic in its interior. Let *a* and *p* be positive real numbers and put

$$\mathcal{Q}(z) = \int\_{-a}^{a} |q(z+t)|^p \, dt \, z$$

It is clear that <sup>Q</sup>(*z*) is continuous for Im*<sup>z</sup>* <sup>≧</sup> 0. Since <sup>|</sup>*q*(*z*)<sup>|</sup> *<sup>p</sup>* is subharmonic for Im*z* > 0 (see [12, p.83]), so is <sup>Q</sup>(*z*).

**Lemma 3.4** Let *q*(*z*) be a function of order of growth≦ *τ* in the half-plane Im*z* ≧ 0 and suppose that the following quantities are both finite:

$$M = \sup\_{-\infty < \chi < \infty} \mathcal{Q}(\mathfrak{x}) and N = \sup\_{y > 0} \mathcal{Q}(iy).$$

Then on this half-plane,

$$
\mathcal{Q}(z) \le \max(M, N).
$$

**Proof.** Since *q*(*z*) is of order of growth≦ *τ*, then there exist positive numbers *A* and *B* such that

$$|q(z)| \le Ae^{B|z|^{\Gamma}} (Imz \ge 0). \tag{5}$$

For each positive number *ε*, define the auxiliary function

$$q\_{\varepsilon}(z) = q(z)e^{-\varepsilon(\lambda(z+a))^{3/2}},\tag{6}$$

where *<sup>λ</sup>* <sup>=</sup> *<sup>e</sup>*−*i<sup>π</sup>*/4. The exponent of *<sup>e</sup>* in (6) has two possible determinations in the half-plane Im*<sup>z</sup>* <sup>&</sup>gt; 0; we choose the one whose real part is negative in the quarter-plane *<sup>x</sup>* <sup>&</sup>gt; <sup>−</sup>*a*, *<sup>y</sup>* <sup>≧</sup> 0. Put

$$\mathcal{Q}\_{\varepsilon}(z) = \int\_{-a}^{a} |q\_{\varepsilon}(z+t)|^{p} \, dt\_{\prime}$$

which is then defined and continuous in the upper half-plane Im*z* ≧ 0, and subharmonic in its interior. A simple calculation involving (5) and (6) shows that in the quarter plane *<sup>x</sup>* <sup>&</sup>gt; <sup>−</sup>*a*, *<sup>y</sup>* <sup>≧</sup> 0,

$$|q\_{\varepsilon}(z)| \le A e^{B|z|^r - \varepsilon \gamma |z + a|^{3/2}},\tag{7}$$

where *<sup>γ</sup>* <sup>=</sup> *cos* <sup>3</sup>*π*/8, and <sup>|</sup>*qε*(*z*)<sup>|</sup> <sup>≦</sup> <sup>|</sup>*q*(*z*)|. Hence

$$\mathcal{Q}\_{\varepsilon}(z) \le \mathcal{Q}(z)(x \ge 0, \, y \ge 0)\_{\prime}$$

and in particular

8 Fourier Transform

then

for some positive number *p*,

[12, p.83]), so is <sup>Q</sup>(*z*).

Then on this half-plane,

that

We have the following version of Plancherel-Pólya theorem. We give the proof which is

**Theorem 3.3 (Plancherel-Pólya).** If *f*(*z*) is an entire function of order of growth≦ *τ* and if

*<sup>p</sup> dx* <sup>≦</sup> *<sup>e</sup>pτ*|*y*<sup>|</sup>

The proof will require two preliminary lemmas. Suppose that *q*(*z*) is a non constant continuous function in the closed upper half-plane, Im*z* ≧ 0, and analytic in its interior.

**Lemma 3.4** Let *q*(*z*) be a function of order of growth≦ *τ* in the half-plane Im*z* ≧ 0 and

<sup>Q</sup>(*z*) <sup>≦</sup> *max*(*M*, *<sup>N</sup>*).

**Proof.** Since *q*(*z*) is of order of growth≦ *τ*, then there exist positive numbers *A* and *B* such

*τ*

−*ε*(*λ*(*z*+*a*))3/2

<sup>|</sup>*q*(*z*)<sup>|</sup> <sup>≦</sup> *AeB*|*z*<sup>|</sup>

*qε*(*z*) = *q*(*z*)*e*

<sup>|</sup>*q*(*<sup>z</sup>* <sup>+</sup> *<sup>t</sup>*)<sup>|</sup>

<sup>Q</sup>(*x*)*andN* <sup>=</sup> sup

*<sup>p</sup> dt*.

*y*>0

<sup>Q</sup>(*iy*).

(*Imz* ≧ 0). (5)

, (6)

 *a* −*a* *<sup>p</sup> dx* < ∞,

 ∞ −∞

<sup>|</sup> *<sup>f</sup>*(*x*)<sup>|</sup>

*<sup>p</sup> dx*.

*<sup>p</sup>* is subharmonic for Im*z* > 0 (see

<sup>|</sup> *<sup>f</sup>*(*x*)<sup>|</sup>

 ∞ −∞

<sup>|</sup> *<sup>f</sup>*(*<sup>x</sup>* <sup>+</sup> *iy*)<sup>|</sup>

<sup>Q</sup>(*z*) =

similar to [11, Th. 2.16] for the sake of completeness.

 ∞ −∞

It is clear that <sup>Q</sup>(*z*) is continuous for Im*<sup>z</sup>* <sup>≧</sup> 0. Since <sup>|</sup>*q*(*z*)<sup>|</sup>

*M* = sup <sup>−</sup>∞<*x*<<sup>∞</sup>

suppose that the following quantities are both finite:

For each positive number *ε*, define the auxiliary function

Let *a* and *p* be positive real numbers and put

$$\mathcal{Q}\_{\varepsilon}(\mathbf{x}) \le M \quad \text{for} \quad \mathbf{x} \ge \mathbf{0} \quad \text{and} \quad \mathcal{Q}\_{\varepsilon}(\mathbf{i}y) \le N \quad \text{for} \quad y \ge \mathbf{0}.$$

Let *z*<sup>0</sup> be a fixed but arbitrary point in the first quadrant. We shall apply the maximum principle to <sup>Q</sup>*ε*(*z*) in the region <sup>Ω</sup> <sup>=</sup> {*<sup>z</sup>* : *Rez* <sup>≧</sup> 0, *Imz* <sup>≧</sup> 0, <sup>|</sup>*z*<sup>|</sup> <sup>≦</sup> *<sup>R</sup>*}, choosing *<sup>R</sup>* large enough so that (i) *<sup>z</sup>*<sup>0</sup> <sup>∈</sup> <sup>Ω</sup>, and (ii) the maximum value of <sup>Q</sup>*ε*(*z*) on <sup>Ω</sup> is not attained on the circular *arc*|*z*<sup>|</sup> <sup>=</sup> *<sup>R</sup>* (this is possible by virtue of (7)). Since <sup>Q</sup>*ε*(*z*) does not reduce to a constant, the maximum value of <sup>Q</sup>*ε*(*z*) on <sup>Ω</sup> must be attained on one of the coordinate axes, and in particular,

$$Q\_{\varepsilon}(z\_0) \le \max(M, N).$$

Now let *<sup>ε</sup>* <sup>→</sup> 0. This establishes the result for the first quadrant; the proof for the second quadrant is similar.

**Lemma 3.5** In addition to the hypotheses of Lemma 3.4, suppose that

$$\lim\_{y \to \infty} q(x + iy) = 0 \tag{8}$$

uniformly in *<sup>x</sup>*, for <sup>−</sup>*<sup>a</sup>* <sup>≦</sup> *<sup>x</sup>* <sup>≦</sup> *<sup>a</sup>*. Then

$$\mathcal{Q}(z) \le \mathcal{M}\_\prime \quad \operatorname{Im} z \ge 0.$$

**Proof.** It is sufficient to show that *<sup>N</sup>* <sup>≦</sup> *<sup>M</sup>*. By virtue of (8), we see that the function <sup>Q</sup>(*iy*) tends to zero as *<sup>y</sup>* <sup>→</sup> <sup>∞</sup>, and so must attain its least upper bound *<sup>N</sup>* for some finite value of *y*, say *y* = *y*0. If *y*<sup>0</sup> = 0, then

$$N = \mathcal{Q}(iy\_0) = \mathcal{Q}(0) \le M.$$

If *<sup>y</sup>*<sup>0</sup> <sup>&</sup>gt; 0, then the maximum principle shows that the least upper bound of <sup>Q</sup>(*z*) in the half-plane Im*z* ≧ 0 cannot be attained at the interior point *z* = *iy*0. Therefore, by Lemma 3.4,

$$N = \mathcal{Q}(i y\_0) < \max(M, N)\_{\prime}$$

and again *N* < *M*.

Theorem 3.3 now follows.

**Proof of Theorem 3.3.** It is sufficient to prove the theorem when *y* > 0 and *f*(*z*) is not identically zero. Let *ε* be a fixed positive number and consider the function

$$q(z) = f(z)e^{i(\pi + \varepsilon)z}.$$

It is easy to see that, for each positive number *<sup>a</sup>*, the functions *<sup>q</sup>*(*z*) and <sup>Q</sup>(*z*) satisfy the conditions Lemmas 3.4 and 3.5. Consequently, for *y* > 0,

$$\mathcal{Q}(iy) \le M < \int\_{-\infty}^{\infty} |q(x)|^p dx.$$

This together with the definitions of *<sup>q</sup>*(*z*) and <sup>Q</sup>(*z*) implies

$$e^{-p(\tau+\varepsilon)y} \int\_{-a}^{a} |f(\mathbf{x}+i\mathbf{y})|^{p} d\mathbf{x} < \int\_{-\infty}^{\infty} |f(\mathbf{x})|^{p} d\mathbf{x}.$$

To get the result, first let *<sup>a</sup>* <sup>→</sup> <sup>∞</sup>, then let *<sup>ε</sup>* <sup>→</sup> 0.

**Proposition 3.6** Let *f*(*z*, *w*) be an entire function of order of growth≦ *τ* and suppose that {*λn*}, {*µn*} are increasing sequences of real numbers such that

$$
\lambda\_{n+1} - \lambda\_n \ge \varepsilon\_1 > 0 \quad \text{and} \quad \mu\_{n+1} - \mu\_n \ge \varepsilon\_2 > 0.
$$

If for some positive number *p*,

$$\sup\_{n} \int\_{-\infty}^{\infty} |f(\mathbf{x}\_{\omega}, \mu\_{n})|^{p} \, d\mathbf{x}\_{z} < \infty \quad \text{and} \quad \sup\_{n} \int\_{-\infty}^{\infty} |f(\lambda\_{n}, \mathbf{x}\_{w})|^{p} \, d\mathbf{x}\_{w} < \infty \tag{9}$$

then

10 Fourier Transform

*y*, say *y* = *y*0. If *y*<sup>0</sup> = 0, then

and again *N* < *M*.

Theorem 3.3 now follows.

**Proof.** It is sufficient to show that *<sup>N</sup>* <sup>≦</sup> *<sup>M</sup>*. By virtue of (8), we see that the function <sup>Q</sup>(*iy*) tends to zero as *<sup>y</sup>* <sup>→</sup> <sup>∞</sup>, and so must attain its least upper bound *<sup>N</sup>* for some finite value of

*<sup>N</sup>* <sup>=</sup> <sup>Q</sup>(*iy*0) = <sup>Q</sup>(0) <sup>≦</sup> *<sup>M</sup>*. If *<sup>y</sup>*<sup>0</sup> <sup>&</sup>gt; 0, then the maximum principle shows that the least upper bound of <sup>Q</sup>(*z*) in the half-plane Im*z* ≧ 0 cannot be attained at the interior point *z* = *iy*0. Therefore, by Lemma 3.4,

*<sup>N</sup>* <sup>=</sup> <sup>Q</sup>(*iy*0) <sup>&</sup>lt; *max*(*M*, *<sup>N</sup>*),

**Proof of Theorem 3.3.** It is sufficient to prove the theorem when *y* > 0 and *f*(*z*) is not

It is easy to see that, for each positive number *<sup>a</sup>*, the functions *<sup>q</sup>*(*z*) and <sup>Q</sup>(*z*) satisfy the

 ∞ −∞

*i*(*τ*+*ε*)*z* .

<sup>|</sup>*q*(*x*)<sup>|</sup>

 ∞ −∞

 ∞ −∞

<sup>|</sup> *<sup>f</sup>*(*λn*, *xw*)<sup>|</sup>

*<sup>p</sup> dxw* < ∞, (9)

<sup>|</sup> *<sup>f</sup>*(*x*)<sup>|</sup>

*pdx*.

*pdx* <

*pdx*.

*q*(*z*) = *f*(*z*)*e*

identically zero. Let *ε* be a fixed positive number and consider the function

<sup>Q</sup>(*iy*) <sup>≦</sup> *<sup>M</sup>* <sup>&</sup>lt;

<sup>|</sup> *<sup>f</sup>*(*<sup>x</sup>* <sup>+</sup> *iy*)<sup>|</sup>

**Proposition 3.6** Let *f*(*z*, *w*) be an entire function of order of growth≦ *τ* and suppose that

*<sup>λ</sup>n*+<sup>1</sup> <sup>−</sup> *<sup>λ</sup><sup>n</sup>* <sup>≧</sup> *<sup>ε</sup>*<sup>1</sup> <sup>&</sup>gt; 0 and *<sup>µ</sup>n*+<sup>1</sup> <sup>−</sup> *<sup>µ</sup><sup>n</sup>* <sup>≧</sup> *<sup>ε</sup>*<sup>2</sup> <sup>&</sup>gt; 0.

*<sup>p</sup> dxz* <sup>&</sup>lt; <sup>∞</sup> and sup*<sup>n</sup>*

conditions Lemmas 3.4 and 3.5. Consequently, for *y* > 0,

This together with the definitions of *<sup>q</sup>*(*z*) and <sup>Q</sup>(*z*) implies

 *a* −*a*

{*λn*}, {*µn*} are increasing sequences of real numbers such that

*e* <sup>−</sup>*p*(*τ*+*ε*)*<sup>y</sup>*

To get the result, first let *<sup>a</sup>* <sup>→</sup> <sup>∞</sup>, then let *<sup>ε</sup>* <sup>→</sup> 0.

If for some positive number *p*,

sup *n*

 ∞ −∞

<sup>|</sup> *<sup>f</sup>*(*xz*, *<sup>µ</sup>n*)<sup>|</sup>

$$\sum\_{n} |f(\lambda\_n, \mu\_n)|^p < \infty.$$

**Proof.** First, using the Plancherel-Pólya Theorem, observe that conditions (9) imply that

$$\sup\_{\mu} \int\_{-\infty}^{\infty} |f(z, \mu\_n)|^p \, d\mathbf{x}\_{\boldsymbol{\Sigma}} \le e^{p\tau \|\boldsymbol{y}\_{\boldsymbol{\Sigma}}\|} \int\_{-\infty}^{\infty} \sup\_{\boldsymbol{n}} |f(\boldsymbol{x}\_{\boldsymbol{\Sigma}}, \mu\_n)|^p \, d\mathbf{x}\_{\boldsymbol{\Sigma}}.$$

and

$$\sup\_{\boldsymbol{\mu}} \int\_{-\infty}^{\infty} |f(\lambda\_{\boldsymbol{\mu}}, \boldsymbol{w})|^{p} \, d\boldsymbol{x}\_{\boldsymbol{w}} \le e^{p\tau |y\_{\boldsymbol{\mu}}|} \int\_{-\infty}^{\infty} \sup\_{\boldsymbol{\mu}} |f(\lambda\_{\boldsymbol{\mu}}, \boldsymbol{x}\_{\boldsymbol{w}})|^{p} \, d\boldsymbol{x}\_{\boldsymbol{w}}.$$

Now since <sup>|</sup> *<sup>f</sup>* <sup>|</sup> *<sup>p</sup>* is plurisubharmonic, the inequality

$$|f(z\_0, w\_0)|^p \le \frac{1}{2\pi} \int\_0^{2\pi} |f((z\_0, w\_0) + (\zeta\_\prime \eta)e^{i\theta})|^p d\theta \tag{10}$$

holds for all values of (*ζ*, *η*). Fix *η* = 0, multiply both sides of (10) by *ζ* and integrate between 0 and *δ*1,

$$\int\_0^{\delta\_1} |f(z\_0, w\_0)|^p \zeta \, d\zeta \le \frac{1}{2\pi} \int\_0^{\delta\_1} \int\_0^{2\pi} |f(z\_0 + \zeta e^{i\theta}, w\_0)|^p \, d\theta \, \zeta \, d\zeta.$$

Then

$$|f(z\_0, w\_0)|^p \le \frac{1}{\pi \delta\_1^2} \iint\_{\Omega\_1} |f(z, w\_0)|^p \, d\mathfrak{x}\_z \, dy\_{z\nu}$$

where <sup>Ω</sup><sup>1</sup> <sup>=</sup> {(*z*, *<sup>w</sup>*0) : <sup>|</sup>*<sup>z</sup>* <sup>−</sup> *<sup>z</sup>*0<sup>|</sup> <sup>≦</sup> *<sup>δ</sup>*1}. Similarly fix *<sup>ζ</sup>* <sup>=</sup> 0, multiply both sides of (10) by *<sup>η</sup>* and integrate between 0 and *δ*2,

$$|f(z\_{0\prime}w\_0)|^p \le \frac{1}{\pi \delta\_2^2} \iint\_{\Omega\_2} |f(z\_{0\prime}w)|^p \,d\chi\_w \,dy\_w\,$$

where <sup>Ω</sup><sup>2</sup> <sup>=</sup> {(*z*0, *<sup>w</sup>*) : <sup>|</sup>*<sup>w</sup>* <sup>−</sup> *<sup>w</sup>*0<sup>|</sup> <sup>≦</sup> *<sup>δ</sup>*2}. Then

$$\begin{split} 2|f(z\_0, w\_0)|^p &\leq \frac{1}{\pi \delta\_1^2} \iint\_{\Omega\_1} |f(z, w\_0)|^p \,d\mathbf{x}\_z \,d\mathbf{y}\_z \\ &+ \frac{1}{\pi \delta\_2^2} \iint\_{\Omega\_2} |f(z\_0, w)|^p \,d\mathbf{x}\_w \,d\mathbf{y}\_w. \end{split}$$

Now let Ω*<sup>n</sup>* <sup>1</sup> <sup>=</sup> {(*λ<sup>n</sup>* <sup>+</sup> *<sup>z</sup>*, *<sup>µ</sup>n*) : <sup>|</sup>*z*<sup>|</sup> <sup>≦</sup> *<sup>δ</sup>*1} and <sup>Ω</sup>*<sup>n</sup>* <sup>2</sup> <sup>=</sup> {(*λn*, *<sup>µ</sup><sup>n</sup>* <sup>+</sup> *<sup>w</sup>*) : <sup>|</sup>*w*<sup>|</sup> <sup>≦</sup> *<sup>δ</sup>*2}, then

2∑*n* <sup>|</sup> *<sup>f</sup>*(*λn*, *<sup>µ</sup>n*)<sup>|</sup> *<sup>p</sup>* <sup>≦</sup> <sup>∑</sup>*<sup>n</sup>* 1 *πδ*2 1 Ω*<sup>n</sup>* 1 <sup>|</sup> *<sup>f</sup>*(*λ<sup>n</sup>* <sup>+</sup> *<sup>z</sup>*, *<sup>µ</sup>n*)<sup>|</sup> *<sup>p</sup> dxz dyz* + 1 *πδ*2 2 Ω*<sup>n</sup>* 2 <sup>|</sup> *<sup>f</sup>*(*λn*, *<sup>µ</sup><sup>n</sup>* <sup>+</sup> *<sup>w</sup>*)<sup>|</sup> *<sup>p</sup> dxw dyw* <sup>≦</sup> <sup>∑</sup>*<sup>n</sup>* 1 *πδ*2 1 *<sup>δ</sup>*<sup>1</sup> −*δ*1 *<sup>δ</sup>*<sup>1</sup> −*δ*1 <sup>|</sup> *<sup>f</sup>*(*λ<sup>n</sup>* <sup>+</sup> *<sup>z</sup>*, *<sup>µ</sup>n*)<sup>|</sup> *<sup>p</sup> dxz dyz* + 1 *πδ*2 2 *<sup>δ</sup>*<sup>2</sup> −*δ*2 *<sup>δ</sup>*<sup>2</sup> −*δ*2 <sup>|</sup> *<sup>f</sup>*(*λn*, *<sup>µ</sup><sup>n</sup>* <sup>+</sup> *<sup>w</sup>*)<sup>|</sup> *<sup>p</sup> dxw dyw* <sup>=</sup> <sup>∑</sup>*<sup>n</sup>* 1 *πδ*2 1 *<sup>δ</sup>*<sup>1</sup> −*δ*1 *<sup>λ</sup>n*+*δ*<sup>1</sup> *<sup>λ</sup><sup>n</sup>*−*δ*<sup>1</sup> <sup>|</sup> *<sup>f</sup>*(*z*, *<sup>µ</sup>n*)<sup>|</sup> *<sup>p</sup> dxz dyz* + 1 *πδ*2 2 *<sup>δ</sup>*<sup>2</sup> −*δ*2 *<sup>µ</sup>n*+*δ*<sup>2</sup> *<sup>µ</sup><sup>n</sup>*−*δ*<sup>2</sup> <sup>|</sup> *<sup>f</sup>*(*λn*, *<sup>w</sup>*)<sup>|</sup> *<sup>p</sup> dxw dyw* .

It is clear that the last expression above is no larger than

$$\begin{split} & \sum\_{n} \left( \frac{1}{\pi \delta\_{1}^{2}} \int\_{-\delta\_{1}}^{\delta\_{1}} \int\_{\lambda\_{n}-\delta\_{1}}^{\lambda\_{n}+\delta\_{1}} \sup\_{n} |f(z, \mu\_{n})|^{p} \, d\mathbf{x}\_{z} \, d\mathbf{y}\_{z} \\ & + \frac{1}{\pi \delta\_{2}^{2}} \int\_{-\delta\_{2}}^{\delta\_{2}} \int\_{\mu\_{n}-\delta\_{2}}^{\mu\_{n}+\delta\_{2}} \sup\_{n} |f(\lambda\_{n}, w)|^{p} \, d\mathbf{x}\_{w} \, d\mathbf{y}\_{w} \right). \end{split}$$

Now for *<sup>δ</sup>*<sup>1</sup> <sup>=</sup> *<sup>ε</sup>*<sup>1</sup> <sup>2</sup> and *<sup>δ</sup>*<sup>2</sup> <sup>=</sup> *<sup>ε</sup>*<sup>2</sup> <sup>2</sup> , the intervals (*λ<sup>n</sup>* <sup>−</sup> *<sup>δ</sup>*1, *<sup>λ</sup><sup>n</sup>* <sup>+</sup> *<sup>δ</sup>*1) are pairwise disjoint, and similarly for the intervals (*µ<sup>n</sup>* <sup>−</sup> *<sup>δ</sup>*2, *<sup>µ</sup><sup>n</sup>* <sup>+</sup> *<sup>δ</sup>*2), thus

$$\begin{split} 2\sum\_{n} |f(\lambda\_{n},\mu\_{n})|^{p} &\leq \frac{1}{\pi\delta\_{1}^{2}} \int\_{-\delta\_{1}}^{\delta\_{1}} \int\_{-\infty}^{\infty} \sup\_{n} |f(z,\mu\_{n})|^{p} \,d\mathbf{x}\_{z} \,d\mathbf{y}\_{z} \\ &+ \frac{1}{\pi\delta\_{2}^{2}} \int\_{-\delta\_{2}}^{\delta\_{2}} \int\_{-\infty}^{\infty} \sup\_{n} |f(\lambda\_{n},w)|^{p} \,d\mathbf{x}\_{w} \,d\mathbf{y}\_{w}. \end{split}$$

We conclude that

$$\begin{split} 2\sum\_{n} |f(\boldsymbol{\lambda}\_{n},\boldsymbol{\mu}\_{n})|^{p} &\leq \frac{1}{\pi\delta\_{1}^{2}} \int\_{-\delta\_{1}}^{\delta\_{1}} \left(e^{p\tau \left|\boldsymbol{y}\_{z}\right|} \int\_{-\infty}^{\infty} \sup\_{n} |f(\boldsymbol{x}\_{z},\boldsymbol{\mu}\_{n})|^{p} \,d\boldsymbol{x}\_{z}\right) d\boldsymbol{y}\_{z} \\ &+ \frac{1}{\pi\delta\_{2}^{2}} \int\_{-\delta\_{2}}^{\delta\_{2}} \left(e^{p\tau \left|\boldsymbol{y}\_{w}\right|} \int\_{-\infty}^{\infty} \sup\_{n} |f(\boldsymbol{\lambda}\_{n},\boldsymbol{x}\_{w})|^{p} \,d\boldsymbol{x}\_{w}\right) d\boldsymbol{y}\_{w} \\ &= B\_{1} \sup\_{n} \int\_{-\infty}^{\infty} |f(\boldsymbol{x}\_{z},\boldsymbol{\mu}\_{n})|^{p} \,d\boldsymbol{x}\_{z} \\ &+ B\_{2} \sup\_{n} \int\_{-\infty}^{\infty} |f(\boldsymbol{\lambda}\_{n},\boldsymbol{x}\_{w})|^{p} \,d\boldsymbol{x}\_{w} < \infty, \end{split}$$

where *B*<sup>1</sup> = *B*1(*p*, *τ*,*ε*1) and *B*<sup>2</sup> = *B*2(*p*, *τ*,*ε*2).

12 Fourier Transform

Now let Ω*<sup>n</sup>*

Now for *<sup>δ</sup>*<sup>1</sup> <sup>=</sup> *<sup>ε</sup>*<sup>1</sup>

We conclude that

2∑*n*

<sup>1</sup> <sup>=</sup> {(*λ<sup>n</sup>* <sup>+</sup> *<sup>z</sup>*, *<sup>µ</sup>n*) : <sup>|</sup>*z*<sup>|</sup> <sup>≦</sup> *<sup>δ</sup>*1} and <sup>Ω</sup>*<sup>n</sup>*

*<sup>p</sup>* <sup>≦</sup> <sup>∑</sup>*<sup>n</sup>*

+ 1 *πδ*2 2

+ 1 *πδ*2 2

+ 1 *πδ*2 2

 *<sup>δ</sup>*<sup>1</sup> −*δ*1

*<sup>p</sup>* <sup>≦</sup> <sup>1</sup> *πδ*2 1

> + 1 *πδ*2 2

> > *<sup>δ</sup>*<sup>1</sup> −*δ*1 *<sup>e</sup>pτ*|*yz* <sup>|</sup>

 *<sup>δ</sup>*<sup>2</sup> −*δ*2 *<sup>e</sup>pτ*|*yw*<sup>|</sup>

 ∞ −∞

 ∞ −∞

*<sup>p</sup>* <sup>≦</sup> <sup>1</sup> *πδ*2 1

> + 1 *πδ*2 2

= *B*<sup>1</sup> sup *n*

+ *B*<sup>2</sup> sup *n*

 *<sup>δ</sup>*<sup>2</sup> −*δ*2

It is clear that the last expression above is no larger than

∑*n* 1 *πδ*2 1

+ 1 *πδ*2 2

<sup>2</sup> and *<sup>δ</sup>*<sup>2</sup> <sup>=</sup> *<sup>ε</sup>*<sup>2</sup>

similarly for the intervals (*µ<sup>n</sup>* <sup>−</sup> *<sup>δ</sup>*2, *<sup>µ</sup><sup>n</sup>* <sup>+</sup> *<sup>δ</sup>*2), thus

<sup>|</sup> *<sup>f</sup>*(*λn*, *<sup>µ</sup>n*)<sup>|</sup>

2∑*n*

<sup>|</sup> *<sup>f</sup>*(*λn*, *<sup>µ</sup>n*)<sup>|</sup>

<sup>≦</sup> <sup>∑</sup>*<sup>n</sup>*

<sup>=</sup> <sup>∑</sup>*<sup>n</sup>*

 1 *πδ*2 1

> Ω*<sup>n</sup>* 2

 1 *πδ*2 1

> *<sup>δ</sup>*<sup>2</sup> −*δ*2

 1 *πδ*2 1

> *<sup>δ</sup>*<sup>2</sup> −*δ*2

 *<sup>λ</sup>n*+*δ*<sup>1</sup> *<sup>λ</sup><sup>n</sup>*−*δ*<sup>1</sup>

 *<sup>µ</sup>n*+*δ*<sup>2</sup> *<sup>µ</sup><sup>n</sup>*−*δ*<sup>2</sup>

 Ω*<sup>n</sup>* 1

 *<sup>δ</sup>*<sup>1</sup> −*δ*1

 *<sup>δ</sup>*<sup>2</sup> −*δ*2

 *<sup>δ</sup>*<sup>1</sup> −*δ*1

 *<sup>µ</sup>n*+*δ*<sup>2</sup> *<sup>µ</sup><sup>n</sup>*−*δ*<sup>2</sup>

> sup *n*

sup *n*

 *<sup>δ</sup>*<sup>1</sup> −*δ*1

 *<sup>δ</sup>*<sup>2</sup> −*δ*2  ∞ −∞ sup *n*

 ∞ −∞ sup *n*

> ∞ −∞ sup *n*

> ∞ −∞ sup *n*

> > *<sup>p</sup> dxz*

*<sup>p</sup> dxw* < ∞,

<sup>|</sup> *<sup>f</sup>*(*xz*, *<sup>µ</sup>n*)<sup>|</sup>

<sup>|</sup> *<sup>f</sup>*(*λn*, *xw*)<sup>|</sup>

<sup>|</sup> *<sup>f</sup>*(*λn*, *<sup>µ</sup>n*)<sup>|</sup>

2∑*n*

<sup>2</sup> <sup>=</sup> {(*λn*, *<sup>µ</sup><sup>n</sup>* <sup>+</sup> *<sup>w</sup>*) : <sup>|</sup>*w*<sup>|</sup> <sup>≦</sup> *<sup>δ</sup>*2}, then

*<sup>p</sup> dxw dyw*

<sup>|</sup> *<sup>f</sup>*(*λ<sup>n</sup>* <sup>+</sup> *<sup>z</sup>*, *<sup>µ</sup>n*)<sup>|</sup>

<sup>|</sup> *<sup>f</sup>*(*z*, *<sup>µ</sup>n*)<sup>|</sup>

*<sup>p</sup> dxz dyz*

*<sup>p</sup> dxw dyw*

<sup>2</sup> , the intervals (*λ<sup>n</sup>* <sup>−</sup> *<sup>δ</sup>*1, *<sup>λ</sup><sup>n</sup>* <sup>+</sup> *<sup>δ</sup>*1) are pairwise disjoint, and

<sup>|</sup> *<sup>f</sup>*(*z*, *<sup>µ</sup>n*)<sup>|</sup>

<sup>|</sup> *<sup>f</sup>*(*λn*, *<sup>w</sup>*)<sup>|</sup>

<sup>|</sup> *<sup>f</sup>*(*xz*, *<sup>µ</sup>n*)<sup>|</sup>

<sup>|</sup> *<sup>f</sup>*(*λn*, *xw*)<sup>|</sup>

*<sup>p</sup> dxz dyz*

*<sup>p</sup> dxw dyw*

*<sup>p</sup> dxz dyz*

 .

*<sup>p</sup> dxw dyw*

 .

*<sup>p</sup> dxz dyz*

*<sup>p</sup> dxw dyw*.

*<sup>p</sup> dxz dyz*

*<sup>p</sup> dxw dyw*

*<sup>p</sup> dxz dyz*

<sup>|</sup> *<sup>f</sup>*(*λ<sup>n</sup>* <sup>+</sup> *<sup>z</sup>*, *<sup>µ</sup>n*)<sup>|</sup>

<sup>|</sup> *<sup>f</sup>*(*λn*, *<sup>µ</sup><sup>n</sup>* <sup>+</sup> *<sup>w</sup>*)<sup>|</sup>

<sup>|</sup> *<sup>f</sup>*(*λn*, *<sup>w</sup>*)<sup>|</sup>

<sup>|</sup> *<sup>f</sup>*(*z*, *<sup>µ</sup>n*)<sup>|</sup>

<sup>|</sup> *<sup>f</sup>*(*λn*, *<sup>w</sup>*)<sup>|</sup>

<sup>|</sup> *<sup>f</sup>*(*λn*, *<sup>µ</sup><sup>n</sup>* <sup>+</sup> *<sup>w</sup>*)<sup>|</sup>

 *<sup>λ</sup>n*+*δ*<sup>1</sup> *<sup>λ</sup><sup>n</sup>*−*δ*<sup>1</sup>

 *<sup>δ</sup>*<sup>1</sup> −*δ*1 **Remark 3.7** In the above proposition, if we replace the conditions (9) by

$$\int\_{-\infty}^{\infty} \int\_{-\infty}^{\infty} |f(\mathbf{x}\_{z}, \mathbf{x}\_{w})|^{p} d\mathbf{x}\_{z} \, d\mathbf{x}\_{w} < \infty$$

the interior integral is finite everywhere except on a null set. If we use Fubini to change the order of integration, we get a null set for the integral against the second variable. If we know that none of *λn*'s and *µn*'s lie in these null sets, the conclusion still holds.

**Theorem 3.8** If {*λn*}*n*∈**<sup>Z</sup>** and {*µn*}*n*∈**<sup>Z</sup>** are separated sequences of real numbers such that 0 ≦ *λ<sup>n</sup>* ≦ 1 and 0 ≦ *µ<sup>n</sup>* ≦ 1 for each *n*, then the Gabor system (2) forms a Bessel sequence in *<sup>L</sup>*2(**R**2). If <sup>∑</sup>*<sup>n</sup>* <sup>|</sup>*cn*<sup>|</sup> <sup>2</sup> < ∞, then the Gabor expansion

$$\sum\_{n} c\_{n} e^{2\pi i \mu\_{n} \mathfrak{F} - \pi (\mathfrak{F} - \lambda\_{n})^{2}}$$

converges in mean to an element of *L*2(**R**2).

**Proof.** If *<sup>φ</sup>* <sup>∈</sup> *<sup>L</sup>*2(**R**2) then the inner product

$$a\_{\mathcal{U}} = \langle \sqrt{2} \, e^{2\pi i \mu\_n \xi - \pi (\xi - \lambda\_n)^2} \, \phi(\xi) \rangle;$$

is just the value *f*(*λn*, *µn*) of the entire function

$$f(z, w) = \sqrt{2} \int\_{\mathbb{R}^2} \varrho(\xi) \, e^{2\pi i w \xi - \pi(\xi - z)^2} \, d\xi; \varrho(\xi) = \overline{\phi(\xi)}.$$

in the Gabor space <sup>G</sup>*<sup>g</sup>* and *<sup>f</sup>* is of order of growth 2. we have

$$\begin{aligned} &\sup\_{n}\int\_{-\infty}^{\infty}|f(\mathbf{x}\_{z},\mu\_{n})|^{p}\,d\mathbf{x}\_{z} \\ &\leq 2^{p/2}M^{p}e^{\pi}\int\_{-\infty}^{\infty}\left[\int\_{\mathbb{R}}e^{-2\pi(\mathbf{x}\_{\xi}-\mathbf{x}\_{z})^{2}}\,d\mathbf{x}\_{\xi}\int\_{\mathbb{R}}e^{-2\pi(|y\_{\xi}|-1)^{2}}\,dy\_{\xi}\right]^{p/2}d\mathbf{x}\_{z} < \infty, \end{aligned}$$

and similarly

$$\sup\_{n} \int\_{-\infty}^{\infty} |f(\lambda\_n, \mathbf{x}\_w)|^p \, d\mathbf{x}\_w < \infty.$$

Therefore *f* satisfies conditions (9) and by Proposition 3.6 we have

$$\sum\_{n} |\langle \sqrt{2} \, e^{2\pi i \mu\_{\hbar} \xi - \pi (\xi - \lambda\_{\hbar})^2}, \phi(\xi) \rangle|^2 = \sum\_{n} |a\_{\hbar}|^2 = \sum\_{n} |f(\lambda\_{n\prime} \mu\_{\hbar})|^2 < \infty.$$

Thus the Gabor system (2) forms a Bessel sequence in *L*2(**R**2). The second part follows from the first by Proposition 2.4.

Paley and Wiener, showed in Theorem XLII of [16] that whenever

$$\lim\_{n \to \pm \infty} (\lambda\_{n+1} - \lambda\_n) = \infty,$$

for a sequence of real numbers {*λn*}, then the exponentials are weakly independent over an arbitrarily short interval: <sup>∑</sup> *aneiλnt* <sup>=</sup> 0 only when all the *an* are zero. The next lemma states a similar statement for the set of complex exponentials replaced by the system (4). Here l.i.m. is used to show the limit in mean-square in *L*2. The proof is almost identical to that of Paley and Wiener.

**Lemma 3.9** Let no *an* vanish, ∑<sup>∞</sup> <sup>−</sup><sup>∞</sup> <sup>|</sup>*an*<sup>|</sup> <sup>2</sup> converge, and let

$$\cdots < \lambda\_{-\mathfrak{n}} < \cdots < \lambda\_{-1} < \lambda\_0 < \lambda\_1 < \cdots < \lambda\_{\mathfrak{n}} < \cdots$$

such that

$$\lim\_{n \to \pm \infty} (\lambda\_{n+1} - \lambda\_n) = \infty.$$

Let

$$f(t) = \text{1.i.} m\_{N \to \infty} \sum\_{-N}^{N} a\_n \, e^{2\pi i \lambda\_n t - \pi (t - \lambda\_n)^2},$$

over every finite range. If *f*(*t*) is equivalent to zero over any interval (*a*, *b*) then *f*(*t*) is equivalent to zero over every interval, and all the *an*'s vanish.

Now we want to show that if the separation of the *λn*'s is great enough then system (4) is a Riesz-Fischer sequence.

**Theorem 3.10** Let {*λn*} be a sequence of real numbers whose differences are nondecreasing and satisfy

$$\sum \frac{1}{(\lambda\_{k+1} - \lambda\_k)^2} < \infty.$$

Then the Gabor system (4) is a Riesz-Fischer sequence in *L*2(**R**).

**Proof.** We adapt the proof of [16, Th. 1]. By the second part of the Proposition 2.4 we have to show that for all finite sequences of scalars {*cn*} and some constant *<sup>m</sup>* <sup>&</sup>gt; 0,

$$m\sum |c\_{n}|^{2} \le \left\| \sum c\_{n} \sqrt[4]{2} e^{2\pi i \lambda\_{n} t - \pi (t - \lambda\_{n})^{2}} \right\|^{2}.\tag{11}$$

Using *c* to denote an *l* <sup>2</sup> sequence {*c*1, *<sup>c</sup>*2, ···}, inequality (11) is the same as

$$\frac{\langle Gc,c\rangle\_{I^2}}{\langle c,c\rangle\_{I^2}} \ge m\_r$$

where the *l* <sup>2</sup> operator *G* is the Gram matrix of the members of the Gabor system (4). It is to be shown that the eigenvalues of finite subsections of *G* are bounded away from zero, which in turn follows from these two conditions:


The first condition is satisfied by Lemma 3.9. To verify condition (2), observe that the entries of *G* = *I* + *M* are

$$\mathbf{g}\_{nm} = \sqrt{2} \int\_{-\infty}^{\infty} e^{2\pi i (\lambda\_{\rm u} - \lambda\_{\rm w})t - \pi (t - \lambda\_{\rm u})^2 - \pi (t - \lambda\_{\rm w})^2} \, dt.$$

Now *M* can be shown to be compact by showing that its Schmidt norm is finite. Since *G* is symmetric, it suffices to show that

$$\sum\_{n=1}^{\infty} \sum\_{m=n+1}^{\infty} g\_{nm}^2 < \infty.$$

The sum is bounded above,

14 Fourier Transform

and Wiener.

such that

Riesz-Fischer sequence.

and satisfy

Let

∑*n* |� √ 2 *e*

the first by Proposition 2.4.

**Lemma 3.9** Let no *an* vanish, ∑<sup>∞</sup>

<sup>2</sup>*πiµnξ*−*π*(*ξ*−*λn*)<sup>2</sup>

Paley and Wiener, showed in Theorem XLII of [16] that whenever

<sup>−</sup><sup>∞</sup> <sup>|</sup>*an*<sup>|</sup>

*<sup>f</sup>*(*t*) = ł.*i*.*m*.*N*→<sup>∞</sup>

equivalent to zero over every interval, and all the *an*'s vanish.

Then the Gabor system (4) is a Riesz-Fischer sequence in *L*2(**R**).

, *<sup>φ</sup>*(*ξ*)�|<sup>2</sup> <sup>=</sup> <sup>∑</sup>*<sup>n</sup>*

Thus the Gabor system (2) forms a Bessel sequence in *L*2(**R**2). The second part follows from

lim *<sup>n</sup>*→±∞(*λn*+<sup>1</sup> <sup>−</sup> *<sup>λ</sup>n*) = <sup>∞</sup>,

for a sequence of real numbers {*λn*}, then the exponentials are weakly independent over an arbitrarily short interval: <sup>∑</sup> *aneiλnt* <sup>=</sup> 0 only when all the *an* are zero. The next lemma states a similar statement for the set of complex exponentials replaced by the system (4). Here l.i.m. is used to show the limit in mean-square in *L*2. The proof is almost identical to that of Paley

<sup>2</sup> converge, and let

··· <sup>&</sup>lt; *<sup>λ</sup>*−*<sup>n</sup>* <sup>&</sup>lt; ··· <sup>&</sup>lt; *<sup>λ</sup>*−<sup>1</sup> <sup>&</sup>lt; *<sup>λ</sup>*<sup>0</sup> <sup>&</sup>lt; *<sup>λ</sup>*<sup>1</sup> <sup>&</sup>lt; ··· <sup>&</sup>lt; *<sup>λ</sup><sup>n</sup>* <sup>&</sup>lt; ···

lim *<sup>n</sup>*→±∞(*λn*+<sup>1</sup> <sup>−</sup> *<sup>λ</sup>n*) = <sup>∞</sup>.

*N* ∑ −*N*

over every finite range. If *f*(*t*) is equivalent to zero over any interval (*a*, *b*) then *f*(*t*) is

Now we want to show that if the separation of the *λn*'s is great enough then system (4) is a

**Theorem 3.10** Let {*λn*} be a sequence of real numbers whose differences are nondecreasing

(*λk*+<sup>1</sup> <sup>−</sup> *<sup>λ</sup>k*)<sup>2</sup> <sup>&</sup>lt; <sup>∞</sup>.

∑ <sup>1</sup>

*an e*

<sup>2</sup>*πiλnt*−*π*(*t*−*λn*)<sup>2</sup>

;


<sup>2</sup> <sup>=</sup> <sup>∑</sup>*<sup>n</sup>*

<sup>|</sup> *<sup>f</sup>*(*λn*, *<sup>µ</sup>n*)<sup>|</sup>

<sup>2</sup> < ∞.

$$\begin{split} \sum\_{n=1}^{\infty} \sum\_{m=n+1}^{\infty} g\_{nm}^{2} &= 2 \sum\_{n=1}^{\infty} \sum\_{m=n+1}^{\infty} \left( \int\_{-\infty}^{\infty} e^{2\pi i (\lambda\_{n} - \lambda\_{m})t - \pi (t - \lambda\_{n})^{2} - \pi (t - \lambda\_{m})^{2}} dt \right)^{2} \\ &\leq 2 \sum\_{n=1}^{\infty} \sum\_{m=n+1}^{\infty} \left( \int\_{-\infty}^{\infty} e^{2\pi i (\lambda\_{n} - \lambda\_{m})t} dt \right)^{2} \\ &= 2 \sum\_{n=1}^{\infty} \sum\_{m=n+1}^{\infty} \lim\_{A \to \infty} \left( \int\_{-A}^{A} e^{2\pi i (\lambda\_{n} - \lambda\_{m})t} dt \right)^{2} \\ &\leq \frac{2}{\pi^{2}} \sum\_{n=1}^{\infty} \sum\_{m=n+1}^{\infty} \frac{1}{(\lambda\_{n} - \lambda\_{m})^{2}} \\ &< \frac{2}{\pi^{2}} \sum\_{n=1}^{\infty} \sum\_{m=n+1}^{\infty} \frac{1}{(\lambda\_{n+1} - \lambda\_{n})^{2} (m - n)^{2}}. \end{split}$$

where (*λ<sup>m</sup>* <sup>−</sup> *<sup>λ</sup>n*) <sup>≦</sup> (*λn*+<sup>1</sup> <sup>−</sup> *<sup>λ</sup>n*)(*<sup>m</sup>* <sup>+</sup> *<sup>n</sup>*) follows from the assumption that differences are nondecreasing. Letting *k* = *m* + *n*, one concludes that

$$\sum\_{n=1}^{\infty} \sum\_{m=n+1}^{\infty} g\_{nm}^2 < \mathbb{C} \sum\_{n=1}^{\infty} \frac{1}{(\lambda\_{n+1} - \lambda\_n)^2} < \infty,$$

establishing the theorem.

**Theorem 3.11** Let

$$f(z) = \int\_{-\infty}^{\infty} \alpha(t) e^{2\pi izt - \pi(t-z)^2} dt.$$

where *<sup>α</sup>* <sup>∈</sup> *<sup>L</sup>*2(**R**). If *<sup>f</sup>*(*µ*) = 0 and *<sup>g</sup>*(*z*) = *<sup>z</sup>* <sup>−</sup> *<sup>λ</sup> <sup>z</sup>* <sup>−</sup> *<sup>µ</sup> <sup>f</sup>*(*z*), then there exists a function *<sup>β</sup>* in *<sup>L</sup>*2(**R**) such that

$$\log(z) = \int\_{-\infty}^{\infty} \beta(t) e^{2\pi i z t - \pi(t-z)^2} dt. \tag{12}$$

Moreover,

$$\beta(t) = a(t) + 2\pi(i+1)(\lambda - \mu)e^{-2\pi i \mu t + \pi(t-\mu)^2} \int\_{-\infty}^{t} a(s)e^{2\pi i \mu s - \pi(s-\mu)^2} ds \tag{13}$$

almost everywhere on **R**.

**Proof.** To motivate the proof, let us suppose that *g*(*z*) is in fact representable in the form (12), and try to deduce (13). If (12) holds, then

$$\frac{1}{z-\mu} \int\_{-\infty}^{\infty} a(t)e^{2\pi i z t - \pi (t-z)^2} dt = \frac{1}{z-\lambda} \int\_{-\infty}^{\infty} \beta(t) e^{2\pi i z t - \pi (t-z)^2} dt.$$

The trick in solving for *β*(*t*) is to transform each of these integrals by first rewriting *<sup>e</sup>*2*πizt*−*π*(*t*−*z*)<sup>2</sup> as

$$e^{2\pi i z t - \pi(t - z)^2} = e^{2\pi i (z - \mu)t + 2\pi i \mu t - \pi(t - \mu)^2 + \pi(z - \mu)(2t - z - \mu)}$$

and then integrating by parts. When this is done, the result is

$$\frac{1}{z-\mu} \int\_{-\infty}^{\infty} \mathfrak{a}(t) e^{2\pi i z t - \pi(t-z)^2} dt = \int\_{-\infty}^{\infty} \mathfrak{a}\_1(t) e^{2\pi i z t - \pi(t-z)^2} dt$$

with

16 Fourier Transform

establishing the theorem.

almost everywhere on **R**.

1 *<sup>z</sup>* <sup>−</sup> *<sup>µ</sup>*

as

*e*

1 *<sup>z</sup>* <sup>−</sup> *<sup>µ</sup>*

**Theorem 3.11** Let

such that

Moreover,

*<sup>e</sup>*2*πizt*−*π*(*t*−*z*)<sup>2</sup>

where (*λ<sup>m</sup>* <sup>−</sup> *<sup>λ</sup>n*) <sup>≦</sup> (*λn*+<sup>1</sup> <sup>−</sup> *<sup>λ</sup>n*)(*<sup>m</sup>* <sup>+</sup> *<sup>n</sup>*) follows from the assumption that differences are

∞ ∑ *n*=1

*α*(*t*)*e*

*β*(*t*)*e*

**Proof.** To motivate the proof, let us suppose that *g*(*z*) is in fact representable in the form

*dt* <sup>=</sup> <sup>1</sup> *<sup>z</sup>* <sup>−</sup> *<sup>λ</sup>*

The trick in solving for *β*(*t*) is to transform each of these integrals by first rewriting

*dt* =

 ∞ −∞

*α*1(*t*)*e*

<sup>2</sup>*πizt*−*π*(*t*−*z*)<sup>2</sup>

1 (*λn*+<sup>1</sup> <sup>−</sup> *<sup>λ</sup>n*)<sup>2</sup> <sup>&</sup>lt; <sup>∞</sup>,

<sup>2</sup>*πizt*−*π*(*t*−*z*)<sup>2</sup>

<sup>2</sup>*πizt*−*π*(*t*−*z*)<sup>2</sup>

−2*πiµt*+*π*(*t*−*µ*)<sup>2</sup> *<sup>t</sup>*

*dt*,

−∞

 ∞ −∞

<sup>2</sup>*πi*(*z*−*µ*)*t*+2*πiµt*−*π*(*t*−*µ*)<sup>2</sup>+*π*(*z*−*µ*)(2*t*−*z*−*µ*)

*β*(*t*)*e*

<sup>2</sup>*πizt*−*π*(*t*−*z*)<sup>2</sup>

*dt*.

.

*dt*,

*α*(*s*)*e*

*<sup>z</sup>* <sup>−</sup> *<sup>µ</sup> <sup>f</sup>*(*z*), then there exists a function *<sup>β</sup>* in *<sup>L</sup>*2(**R**)

*dt*. (12)

*ds* (13)

<sup>2</sup>*πiµs*−*π*(*s*−*µ*)<sup>2</sup>

nondecreasing. Letting *k* = *m* + *n*, one concludes that

∞ ∑ *n*=1

where *<sup>α</sup>* <sup>∈</sup> *<sup>L</sup>*2(**R**). If *<sup>f</sup>*(*µ*) = 0 and *<sup>g</sup>*(*z*) = *<sup>z</sup>* <sup>−</sup> *<sup>λ</sup>*

*<sup>β</sup>*(*t*) = *<sup>α</sup>*(*t*) + <sup>2</sup>*π*(*<sup>i</sup>* <sup>+</sup> <sup>1</sup>)(*<sup>λ</sup>* <sup>−</sup> *<sup>µ</sup>*)*<sup>e</sup>*

(12), and try to deduce (13). If (12) holds, then

 ∞ −∞

*α*(*t*)*e*

<sup>2</sup>*πizt*−*π*(*t*−*z*)<sup>2</sup>

 ∞ −∞ <sup>2</sup>*πizt*−*π*(*t*−*z*)<sup>2</sup>

= *e*

<sup>2</sup>*πizt*−*π*(*t*−*z*)<sup>2</sup>

and then integrating by parts. When this is done, the result is

*α*(*t*)*e*

∞ ∑ *m*=*n*+1

*f*(*z*) =

*g*(*z*) =

 ∞ −∞

 ∞ −∞

*g*2 *nm* < *<sup>C</sup>*

$$\alpha\_1(t) = -2(i+1)\pi e^{-2\pi i \mu t + \pi(t-\mu)^2} \int\_{-\infty}^t \alpha(s) e^{2\pi i \mu s - \pi(s-\mu)^2} \, ds \,\mu$$

and

$$\frac{1}{z-\lambda} \int\_{-\infty}^{\infty} \beta(t) e^{2\pi i z t - \pi(t-z)^2} dt = \int\_{-\infty}^{\infty} \beta\_1(t) e^{2\pi i z t - \pi(t-z)^2} dt$$

with

$$\beta\_1(t) = -2(i+1)\pi e^{-2\pi i\lambda t + \pi(t-\lambda)^2} \int\_{-\infty}^t \beta(s) e^{2\pi i\lambda s - \pi(s-\lambda)^2} ds.$$

It follows that *α*1(*t*) = *β*1(*t*) almost everywhere on **R**, and so

$$\begin{aligned} \, \, \_t e^{2\pi i(\lambda-\mu)t + \pi(\lambda-\mu)(2t-(\lambda+\mu))} \int\_{-\infty}^t \mathfrak{a}(s) e^{2\pi i \mu s - \pi(s-\mu)^2} \, ds \\ &= \int\_{-\infty}^t \mathfrak{f}(s) e^{2\pi i \lambda s - \pi(s-\lambda)^2} \, ds. \end{aligned}$$

To obtain (13), differentiate both sides of this equation with respect to *t*. Now simply observe that all of the above steps are reversible, that is *<sup>β</sup>* <sup>∈</sup> *<sup>L</sup>*2(**R**).

**Remark 3.12** A similar result holds when *f* is of the form

$$f(z) = \int\_{-\infty}^{\infty} e^{2\pi izt - \pi(t-z)^2} d\alpha(t),$$

and *α* is of bounded variation on **R**, only now

$$\mathfrak{g}(z) = \int\_{-\infty}^{\infty} e^{2\pi izt - \pi(t-z)^2} d\beta(t).$$

with

$$d\beta(t) = d\mathfrak{a}(t) + 2\pi i (\lambda - \mu) e^{-2\pi i \mu t + \pi (t - \mu)^2} \int\_{-\infty}^{t} e^{2\pi i \mu s - \pi (s - \mu)^2} d\mathfrak{a}(s).$$

**Corollary 3.13** The completeness of system (4) is unaffected if one *λ<sup>n</sup>* is replaced by another number.

Nowak [18] showed that the deficit of the regular Gabor system generated by *<sup>h</sup>* <sup>∈</sup> *<sup>L</sup>*2(**R***d*) and *a*, *b* > 0 is either zero or infinite if the system is a Bessel sequence in *L*2(**R***d*). The next result on the deficit of the irregular Gabor system (4) is proved as in [19, Th. 4.6]. Here we give the proof for the sake of completeness.

**Theorem 3.14** If {*λn*} is a separated sequence of real numbers such that

$$
\lambda\_{n+1} - \lambda\_n > 1; \ (n = 0, \pm 1, \pm 2, \dotsb);
$$

then the Gabor system (4) has infinite deficiency in *L*2(**R**).

**Proof.** Let *N* be a fixed but arbitrary positive integer. If *K* is large enough, then we can replace

$$
\lambda\_{0\prime} \lambda\_{1\prime} \cdots \lambda\_{\prime} \lambda\_{\mathcal{K}}
$$

by

*<sup>µ</sup>*0, *<sup>µ</sup>*1, ··· , *<sup>µ</sup>K*+*N*+1,

so that the resulting sequence, relabeled {*µn*}, satisfies

$$\inf\_{n} (\mu\_{n+1} - \mu\_n) > 1.$$

By Theorem 3.10 there is a function *<sup>ϕ</sup>* <sup>∈</sup> *<sup>L</sup>*2(**R**) such that

$$\int\_{\mathbb{R}} \varphi(t) \sqrt[4]{2} \, e^{-2\pi i \mu\_n t - \pi (t - \mu\_n)^2} \, dt = \begin{cases} 1 & \text{if } n = 0, \\ 0 & \text{if } n \neq 0 \end{cases}.$$

Thus the system

$$\left\{ \sqrt[4]{2} \, e^{2\pi i \mu\_n t - \pi (t - \mu\_n)^2} : n \neq 0 \right\}$$

is incomplete in *L*2(**R**), and we conclude by the above corollary that the deficiency of the system in *L*2(**R**) is at least *N*.

## **4. Stability**

In this section we study stability of sampling sets in Gabor spaces. Here we let <sup>G</sup>*<sup>h</sup>* to be the Gabor space of a Gabor window *<sup>h</sup>* <sup>∈</sup> *<sup>L</sup>*2(**R***d*), and <sup>Λ</sup> be a discrete set in **<sup>C</sup>***d*.

**Proposition 4.1** [17, Cor. 3.2.3] (Inversion formula for the Gabor transform) Let *<sup>h</sup>*, *<sup>γ</sup>* <sup>∈</sup> *<sup>L</sup>*2(**R***d*) be such that �*h*, *<sup>γ</sup>*� �<sup>=</sup> 0, and we consider *Vh <sup>f</sup>*(*z*) = �*<sup>f</sup>* , *hz*�, for every *<sup>f</sup>* <sup>∈</sup> *<sup>L</sup>*2(**R***d*). Then it is fulfilled that *Vh <sup>f</sup>* <sup>∈</sup> *<sup>L</sup>*2(**C***d*). Moreover we have inversion formula given by:

$$f(t) = \frac{1}{\langle h, \gamma \rangle} \int\_{\mathbb{C}^d} V\_h f(z) \, M\_y T\_x \gamma(t) \, dy \, dx; \quad z = x + iy \in \mathbb{C}^d.$$

The image of *L*2(**R***d*) under the Gabor transform with the window *h*, forms a reproducing kernel Hilbert space

$$\mathcal{G}\_h = \{ V\_h f \colon f \in L^2(\mathbb{R}^d) \}$$

(a closed subspace of *L*2(**C***d*)) which is called *Gabor space* or *model space*.

The following result is proved for *d* = 1 in [19, Prop. 1.29], the proof given here is based on [17].

**Proposition 4.2** The Gabor space <sup>G</sup>*<sup>h</sup>* of a Gabor window *<sup>h</sup>* <sup>∈</sup> *<sup>L</sup>*2(**R***d*) is a Hilbert subspace of *L*2(**C***d*) that is characterized for the following reproducing kernel:

$$k\_h(z, z\_0) = k(z, z\_0) = k\_{z\_0}(z) = \langle h\_{z\_0}, h\_z \rangle.$$

That is, *<sup>F</sup>* ∈ G*<sup>h</sup>* if and only if

$$F \in L^2(\mathbb{C}^d)$$

and

18 Fourier Transform

replace

Thus the system

**4. Stability**

system in *L*2(**R**) is at least *N*.

by

**Theorem 3.14** If {*λn*} is a separated sequence of real numbers such that

then the Gabor system (4) has infinite deficiency in *L*2(**R**).

so that the resulting sequence, relabeled {*µn*}, satisfies

By Theorem 3.10 there is a function *<sup>ϕ</sup>* <sup>∈</sup> *<sup>L</sup>*2(**R**) such that

 **R** *ϕ*(*t*) √<sup>4</sup> 2 *e*

*<sup>f</sup>*(*t*) = <sup>1</sup>

�*h*, *<sup>γ</sup>*�

 **C***d*

*<sup>λ</sup>n*+<sup>1</sup> <sup>−</sup> *<sup>λ</sup><sup>n</sup>* <sup>&</sup>gt; 1; (*<sup>n</sup>* <sup>=</sup> 0, <sup>±</sup>1, <sup>±</sup>2, ···)

**Proof.** Let *N* be a fixed but arbitrary positive integer. If *K* is large enough, then we can

*<sup>λ</sup>*0, *<sup>λ</sup>*1, ··· , *<sup>λ</sup><sup>K</sup>*

*<sup>µ</sup>*0, *<sup>µ</sup>*1, ··· , *<sup>µ</sup>K*+*N*+1,

*<sup>n</sup>* (*µn*+<sup>1</sup> <sup>−</sup> *<sup>µ</sup>n*) <sup>&</sup>gt; 1.

*dt* =

: *<sup>n</sup>* �<sup>=</sup> <sup>0</sup> 

*Vh <sup>f</sup>*(*z*) *MyTxγ*(*t*) *dy dx*; *<sup>z</sup>* <sup>=</sup> *<sup>x</sup>* <sup>+</sup> *iy* <sup>∈</sup> **<sup>C</sup>***d*.

 1 if *n* = 0, 0 if *<sup>n</sup>* �<sup>=</sup> <sup>0</sup> .

−2*πiµnt*−*π*(*t*−*µn*)<sup>2</sup>

<sup>2</sup>*πiµnt*−*π*(*t*−*µn*)<sup>2</sup>

is incomplete in *L*2(**R**), and we conclude by the above corollary that the deficiency of the

In this section we study stability of sampling sets in Gabor spaces. Here we let <sup>G</sup>*<sup>h</sup>* to be the

**Proposition 4.1** [17, Cor. 3.2.3] (Inversion formula for the Gabor transform) Let *<sup>h</sup>*, *<sup>γ</sup>* <sup>∈</sup> *<sup>L</sup>*2(**R***d*) be such that �*h*, *<sup>γ</sup>*� �<sup>=</sup> 0, and we consider *Vh <sup>f</sup>*(*z*) = �*<sup>f</sup>* , *hz*�, for every *<sup>f</sup>* <sup>∈</sup> *<sup>L</sup>*2(**R***d*). Then it is

inf

<sup>√</sup><sup>4</sup> <sup>2</sup> *<sup>e</sup>*

Gabor space of a Gabor window *<sup>h</sup>* <sup>∈</sup> *<sup>L</sup>*2(**R***d*), and <sup>Λ</sup> be a discrete set in **<sup>C</sup>***d*.

fulfilled that *Vh <sup>f</sup>* <sup>∈</sup> *<sup>L</sup>*2(**C***d*). Moreover we have inversion formula given by:

$$F(z\_0) = \int\_{\mathbb{C}^d} F(z) \, \overline{k(z, z\_0)} \, dx \, dy. \tag{14}$$

**Proof.** We introduced the inversion formula for the Gabor transform in Proposition 4.1. Without loss of generality, we may assume that �*h*� <sup>=</sup> 1. Now for *<sup>z</sup>*<sup>0</sup> <sup>=</sup> *<sup>x</sup>*<sup>0</sup> <sup>+</sup> *iy*<sup>0</sup> <sup>∈</sup> **<sup>C</sup>***<sup>d</sup>* we have

$$\begin{aligned} V\_h f(z\_0) &= \int\_{\mathbb{R}^d} f(t) \, \overline{M\_{y\_0} T\_{\mathcal{X}\_0} h(t)} \, dt \\ &= \int\_{\mathbb{R}^d} \left( \int\_{\mathbb{C}^d} V\_h f(z) M\_{\mathcal{Y}} T\_{\mathcal{X}} h(t) d\mathfrak{x} \, dy \right) \overline{M\_{y\_0} T\_{\mathcal{X}\_0} h(t)} \, dt \end{aligned}$$

where *z* = *x* + *iy*. Switching the integrals we have

$$\begin{aligned} V\_h f(z\_0) &= \int\_{\mathbb{C}^d} V\_h f(z) \left( \int\_{\mathbb{R}^d} M\_{\mathcal{Y}} T\_{\ge} h(t) \, \overline{M\_{\mathcal{Y}0} T\_{\ge} h(t)} \, dt \right) d\mathbf{x} \, dy \\ &= \int\_{\mathbb{C}^d} V\_h f(z) \left( \int\_{\mathbb{R}^d} h\_z(t) \overline{h\_{z\_0}(t)} \, dt \right) d\mathbf{x} \, dy \, \end{aligned}$$

that is,

$$\begin{aligned} F(z\_0) &= \int\_{\mathbb{C}^d} F(z) \langle h\_{\overline{z}, \overline{h}\_{\overline{z\_0}}} \rangle d\mathbb{x} \, dy \\ &= \int\_{\mathbb{C}^d} F(z) \overline{k(z, z\_0)} d\mathbb{x} \, dy. \end{aligned}$$

Therefore *k* replays all the functions of the space, and as it belongs to this space, it is its reproducing kernel.

For *<sup>x</sup>*, *<sup>y</sup>* <sup>∈</sup> **<sup>R</sup>***<sup>d</sup>* we recall *Tx* describes a translation by *<sup>x</sup>* also called a time shift and *My* <sup>a</sup> modulation by *y* also called a frequency shift. So the operators of the form *MyTx* or *TxMy* are known as time-frequency shifts. They satisfy the commutation relations

$$\begin{aligned} T\_{\boldsymbol{x}}M\_{\mathcal{Y}}f(t) &= (M\_{\mathcal{Y}}f)(t-\boldsymbol{x})\\ &= e^{2\pi i \boldsymbol{y}.(t-\boldsymbol{x})}f(t-\boldsymbol{x})\\ &= e^{-2\pi i \boldsymbol{y}.\boldsymbol{x}}M\_{\mathcal{Y}}T\_{\boldsymbol{x}}f(t) \end{aligned}$$

Then we have

$$
\begin{aligned}
\langle h\_{z}, h\_{z\_{0}}\rangle &= \langle \mathcal{M}\_{y}T\_{x}h, \mathcal{M}\_{y\_{0}}T\_{x\_{0}}h\rangle \\ &= \langle h, T\_{-x}\mathcal{M}\_{y\_{0}-y}T\_{x\_{0}}h\rangle \\ &= \langle h, e^{2\pi i x.(y\_{0}-y)}\mathcal{M}\_{y\_{0}-y}T\_{x\_{0}-x}h\rangle \\ &= e^{2\pi i x.(y-y\_{0})}\langle h, \mathcal{M}\_{y\_{0}-y}T\_{x\_{0}-x}h\rangle \\ &= e^{2\pi i x.(y-y\_{0})}\langle h, h\_{z\_{0}-z}\rangle
\end{aligned}
$$

In terms of *kh*(*z*) = *kh*(*z*, 0) = �*h*, *hz*� one has

$$\begin{aligned} k\_{\hbar}(z, z\_0) &= \overline{\langle h\_{z}, h\_{z\_0} \rangle} = \overline{\overline{e^{2\pi i x.(y - y\_0)} \langle h\_{\hbar} h\_{z\_0 - z} \rangle}} \\ &= \overline{\overline{e^{2\pi i x.(y - y\_0)} k\_{\hbar}(z\_0 - z)}} \end{aligned}$$

and hence the reproduction formula (14) takes the form

$$F(z\_0) = \int\_{\mathbb{C}^d} F(z)e^{2\pi i x.(y-y\_0)} k\_h(z\_0 - z) dx \, dy$$

Using this notations we can deduce that

$$\begin{aligned} V\_{\hbar} f\_{z\_0}(z) &= \langle f\_{z\_0}, h\_z \rangle = e^{2\pi i \mathbf{x}\_0 \cdot (y\_0 - y)} \langle f, h\_{z - z\_0} \rangle \\ &= e^{2\pi i \mathbf{x}\_0 \cdot (y\_0 - y)} V\_{\hbar} f(z - z\_0). \end{aligned}$$

In this way, to be consistent with the notation and the definition of the transform, we have to define the translations in **<sup>C</sup>***<sup>d</sup>* of a function *<sup>F</sup>* ∈ G*<sup>h</sup>* (or in *<sup>L</sup>*2(**C***d*) in a general way) as:

$$F\_{z\_0}(z) = e^{2\pi i \mathbf{x}\_0 \cdot (y\_0 - y)} F(z - z\_0).$$

10.5772/60034

It is necessary to observe that these translations do not coincide in general with the usual translation of **C***d*. But if we look at the function, then we have

$$|F\_{z\_0}(z)| = |F(z - z\_0)|.$$

Since in general the function *<sup>F</sup>*(*<sup>z</sup>* <sup>−</sup> *<sup>z</sup>*0) can not belong to <sup>G</sup>*h*. Taking this into account we can write the reproduction formula in a bit more compact way

$$F(z\_0) = \int\_{\mathbb{C}^d} F(z)k\_z(z\_0)dx\,dy$$

The Gabor space has certain good continuity properties. More precisely, the functions of the space will be uniformly continuous. For *<sup>F</sup>* ∈ G*h*, since *<sup>F</sup>* is defined as a definite integral, it is uniformly continuous with respect to the free variable of the integrand, i.e. for each *ε* > 0 there exists *<sup>δ</sup>* <sup>&</sup>gt; 0 such that if <sup>|</sup>*z*<sup>1</sup> <sup>−</sup> *<sup>z</sup>*2<sup>|</sup> <sup>&</sup>lt; *<sup>δ</sup>* by using triangle inequality we have

$$\left| |F(z\_1)| - |F(z\_2)| \right| \le |F(z\_1) - F(z\_2)| < \varepsilon$$

Ascensi [19] formalized this idea in the case *d* = 1 with the following result.

**Proposition 4.3** [19, Prop. 5.1] Let <sup>G</sup>*<sup>h</sup>* be the Gabor space of normalized Gabor window *<sup>h</sup>* <sup>∈</sup> *<sup>L</sup>*2(**R***d*). Then, given *<sup>ε</sup>* there exists *<sup>δ</sup>* such that if <sup>|</sup>*z*<sup>1</sup> <sup>−</sup> *<sup>z</sup>*2<sup>|</sup> <sup>&</sup>lt; *<sup>δ</sup>*, then for every *<sup>F</sup>* ∈ G*<sup>h</sup>*

$$F(z) = \langle f\_\prime h\_z \rangle$$

it is fulfilled that

20 Fourier Transform

reproducing kernel.

Then we have

Therefore *k* replays all the functions of the space, and as it belongs to this space, it is its

For *<sup>x</sup>*, *<sup>y</sup>* <sup>∈</sup> **<sup>R</sup>***<sup>d</sup>* we recall *Tx* describes a translation by *<sup>x</sup>* also called a time shift and *My* <sup>a</sup> modulation by *y* also called a frequency shift. So the operators of the form *MyTx* or *TxMy*

> *TxMy <sup>f</sup>*(*t*)=(*My <sup>f</sup>*)(*<sup>t</sup>* <sup>−</sup> *<sup>x</sup>*) = *e*

> > = *e*

�*hz*, *hz*<sup>0</sup> � <sup>=</sup> �*MyTxh*, *My*0*Tx*<sup>0</sup> *<sup>h</sup>*�

<sup>=</sup> �*h*,*<sup>e</sup>*

= *e*

= *e*

In terms of *kh*(*z*) = *kh*(*z*, 0) = �*h*, *hz*� one has

Using this notations we can deduce that

and hence the reproduction formula (14) takes the form

*F*(*z*0) =

 **C***d F*(*z*)*e*

*Vh fz*<sup>0</sup> (*z*) = �*fz*<sup>0</sup> , *hz*� <sup>=</sup> *<sup>e</sup>*

*Fz*<sup>0</sup> (*z*) = *e*

<sup>=</sup> �*h*, *<sup>T</sup>*−*xMy*<sup>0</sup>−*yTx*<sup>0</sup> *<sup>h</sup>*�

*kh*(*z*, *<sup>z</sup>*0) = �*hz*, *hz*<sup>0</sup> � <sup>=</sup> *<sup>e</sup>*2*πix*.(*y*−*y*0)�*h*, *hz*<sup>0</sup>−*z*�

<sup>2</sup>*πix*.(*y*−*y*0)

= *e*

define the translations in **<sup>C</sup>***<sup>d</sup>* of a function *<sup>F</sup>* ∈ G*<sup>h</sup>* (or in *<sup>L</sup>*2(**C***d*) in a general way) as:

In this way, to be consistent with the notation and the definition of the transform, we have to

<sup>2</sup>*πix*0.(*y*<sup>0</sup>−*y*)

<sup>2</sup>*πix*0.(*y*<sup>0</sup>−*y*)

<sup>2</sup>*πix*0.(*y*<sup>0</sup>−*y*)*F*(*<sup>z</sup>* <sup>−</sup> *<sup>z</sup>*0).

<sup>2</sup>*πix*.(*y*−*y*0)

<sup>2</sup>*πix*.(*y*−*y*0)

<sup>2</sup>*πiy*.(*t*−*x*) *<sup>f</sup>*(*<sup>t</sup>* <sup>−</sup> *<sup>x</sup>*)

<sup>−</sup>2*πiy*.*xMyTx <sup>f</sup>*(*t*)

<sup>2</sup>*πix*.(*y*<sup>0</sup>−*y*)*My*<sup>0</sup>−*yTx*<sup>0</sup>−*xh*�

�*h*, *hz*<sup>0</sup>−*z*�

�*h*, *My*<sup>0</sup>−*yTx*<sup>0</sup>−*xh*�

<sup>=</sup> *<sup>e</sup>*2*πix*.(*y*−*y*0)*kh*(*z*<sup>0</sup> <sup>−</sup> *<sup>z</sup>*)

*kh*(*z*<sup>0</sup> <sup>−</sup> *<sup>z</sup>*)*dx dy*

�*<sup>f</sup>* , *hz*−*z*<sup>0</sup> �

*Vh <sup>f</sup>*(*<sup>z</sup>* <sup>−</sup> *<sup>z</sup>*0).

are known as time-frequency shifts. They satisfy the commutation relations

$$\left| |F(z\_1)| - |F(z\_2)| \right| < \|F\| \varepsilon = \|f\| \varepsilon.$$

A good description of the Gabor space is most convenient if there are some complete characterizations. The best situation occurs when for some analyzing function the Gabor space is a space of holomorphic functions. The most important example and the only possible one is the Gaussian function, for which the Gabor space can be identified with the Fock space, in which the sampling and interpolation sets are completely characterized [20]. The following assertion is proved for *d* = 1 in [7], we do not know if the same holds in higher dimensions.

**Problem 4.4** Consider the Gabor space with a Gabor window *<sup>h</sup>* <sup>∈</sup> *<sup>L</sup>*2(**R***d*)

$$\mathcal{G}\_h = \left\{ F(z) = \int\_{\mathbb{R}^d} f(t) e^{-2\pi i y \cdot t} \overline{h(t-x)} \, dt, \, f \in L^2(\mathbb{R}^d) \right\}.$$

Then this space is a space of antiholomorphic functions (i.e., *<sup>F</sup>*(*x*, <sup>−</sup>*y*) is holomorphic), modulo a multiplication by a weight, if and only if *h* is a time-frequency translation of the Gaussian function.

As the space in that we will work is formed by continuous functions and it has reproducing kernel, we can sample the functions at any point. Given a set of points of **C***<sup>d</sup>* we can consider, for each *<sup>F</sup>* ∈ G*h*, the succession of values that *<sup>F</sup>* takes in this set.

**Definition 4.5** A discrete set <sup>Λ</sup> <sup>=</sup> {*zj*}*j*∈**<sup>Z</sup>** in **<sup>C</sup>***<sup>d</sup>* is said to be a *sampling set* for <sup>G</sup>*<sup>h</sup>* if there are constants *A*, *B* > 0 such that

$$A\|F\|^2 \le \sum\_{j \in \mathbb{Z}} |F(z\_j)|^2 \le B\|F\|^2 F \in \mathcal{G}\_\hbar.$$

These sets are very important since they correspond with frames. We give some properties of Gabor space and sampling set in the case *d* > 1. The case *d* = 1 was considered by Ascensi and Bruna in [7]. The proofs are essentially the same (with little changes in certain cases). Recall that <sup>G</sup>*<sup>h</sup>* <sup>=</sup> {*Vh <sup>f</sup>* : *<sup>f</sup>* <sup>∈</sup> *<sup>L</sup>*2(**R***d*)} is the Gabor space. Here we assume that �*h*� <sup>=</sup> 1.

**Proposition 4.6** If <sup>Λ</sup> <sup>=</sup> {*zj*}*j*∈**<sup>N</sup>** is a sampling set for <sup>G</sup>*<sup>h</sup>* then <sup>Λ</sup> is a relatively separated set.

**Proof.** The proof is exactly similar to the proof of [7, Prop. 3.1].

**Definition 4.7** Given a continuous function *F* defined in **C***<sup>d</sup>* we define its *local maximal function* as:

$$MF(z) = \sup\_{|w-z|<1} |F(w)|$$

**Lemma 4.8** Let <sup>Λ</sup> <sup>=</sup> {*λj*}*j*∈**<sup>Z</sup>** be a separated set with separation constant *<sup>ε</sup>*. Then

$$\sum\_{\lambda \in \Lambda} |k(\lambda)| < \frac{1}{c\varepsilon^{2d}} ||Mk||\_{1/\sigma}$$

where *c* = *m*(*B*(0, 1)).

**Proof.** We suppose without loss of generality that 1 < *ε* < 2. Then using sub-mean-value inequality

$$\begin{split} \sum\_{\lambda \in \Lambda} |k(\lambda)| &\leq \sum\_{\lambda \in \Lambda} \frac{1}{|B(\lambda, \varepsilon)|} \int\_{B(\lambda, \varepsilon)} |k(z)| \, dm(z) \\ &\leq \sum\_{\lambda \in \Lambda} \frac{1}{|B(\lambda, \varepsilon)|} \int\_{B(\lambda, \varepsilon)} Mk(z) \, dm(z) \\ &\leq \sum\_{\lambda \in \Lambda} \frac{1}{|B(\lambda, \varepsilon)|} \int\_{\mathbb{C}^d} Mk(z) \, dm(z) .\end{split}$$

Since by hypothesis those balls are disjoint, then

22 Fourier Transform

as:

where *c* = *m*(*B*(0, 1)).

inequality

constants *A*, *B* > 0 such that

As the space in that we will work is formed by continuous functions and it has reproducing kernel, we can sample the functions at any point. Given a set of points of **C***<sup>d</sup>* we can consider,

**Definition 4.5** A discrete set <sup>Λ</sup> <sup>=</sup> {*zj*}*j*∈**<sup>Z</sup>** in **<sup>C</sup>***<sup>d</sup>* is said to be a *sampling set* for <sup>G</sup>*<sup>h</sup>* if there are

These sets are very important since they correspond with frames. We give some properties of Gabor space and sampling set in the case *d* > 1. The case *d* = 1 was considered by Ascensi and Bruna in [7]. The proofs are essentially the same (with little changes in certain cases). Recall that <sup>G</sup>*<sup>h</sup>* <sup>=</sup> {*Vh <sup>f</sup>* : *<sup>f</sup>* <sup>∈</sup> *<sup>L</sup>*2(**R***d*)} is the Gabor space. Here we assume that �*h*� <sup>=</sup> 1.

**Proposition 4.6** If <sup>Λ</sup> <sup>=</sup> {*zj*}*j*∈**<sup>N</sup>** is a sampling set for <sup>G</sup>*<sup>h</sup>* then <sup>Λ</sup> is a relatively separated set.

**Definition 4.7** Given a continuous function *F* defined in **C***<sup>d</sup>* we define its *local maximal function*

<sup>|</sup>*w*−*z*|<<sup>1</sup>

1 *<sup>c</sup>ε*2*<sup>d</sup>* �*Mk*�1,

**Proof.** We suppose without loss of generality that 1 < *ε* < 2. Then using sub-mean-value

1 <sup>|</sup>*B*(*λ*,*ε*)<sup>|</sup>

1 <sup>|</sup>*B*(*λ*,*ε*)<sup>|</sup>

1 <sup>|</sup>*B*(*λ*,*ε*)<sup>|</sup>  *B*(*λ*,*ε*)

 *B*(*λ*,*ε*)

 **C***d* <sup>|</sup>*k*(*z*)<sup>|</sup> *dm*(*z*)

*Mk*(*z*) *dm*(*z*)

*Mk*(*z*) *dm*(*z*).

<sup>|</sup>*F*(*w*)<sup>|</sup>

*MF*(*z*) = sup

**Lemma 4.8** Let <sup>Λ</sup> <sup>=</sup> {*λj*}*j*∈**<sup>Z</sup>** be a separated set with separation constant *<sup>ε</sup>*. Then

<sup>|</sup>*k*(*λ*)<sup>|</sup> <sup>&</sup>lt;

∑ *<sup>λ</sup>*∈<sup>Λ</sup>

<sup>|</sup>*k*(*λ*)<sup>|</sup> <sup>≦</sup> ∑

*<sup>λ</sup>*∈<sup>Λ</sup>

≦ ∑ *<sup>λ</sup>*∈<sup>Λ</sup>

≦ ∑ *<sup>λ</sup>*∈<sup>Λ</sup>

∑ *<sup>λ</sup>*∈<sup>Λ</sup> <sup>2</sup> <sup>≦</sup> *<sup>B</sup>*�*F*�2*<sup>F</sup>* ∈ G*h*.

<sup>|</sup>*F*(*zj*)<sup>|</sup>

for each *<sup>F</sup>* ∈ G*h*, the succession of values that *<sup>F</sup>* takes in this set.

*<sup>A</sup>*�*F*�<sup>2</sup> <sup>≦</sup> ∑

**Proof.** The proof is exactly similar to the proof of [7, Prop. 3.1].

*j*∈**Z**

$$\sum\_{\lambda \in \Lambda} |k(\lambda)| < \frac{1}{c\varepsilon^{2d}} ||Mk||\_1.$$

**Proposition 4.9** If Λ is a separated set, there exists *B* > 0 such that

$$\sum\_{\lambda \in \Lambda} |F(\lambda)|^2 \le \mathcal{B} \|F\|^2; \quad F \in \mathcal{G}\_{\hbar}.$$

**Proof.** Calculating directly we have that

.

$$\begin{split} \sum\_{\lambda \in \Lambda} |F(\lambda)|^2 &= \sum\_{\lambda \in \Lambda} \left| \int\_{\mathbb{C}^d} F(z) \overline{k\_\lambda(z)} \, dm(z) \right|^2 \\ &\leq \sum\_{\lambda \in \Lambda} \left( \int\_{\mathbb{C}^d} |F(z)|^2 |k\_\lambda(z)| \, dm(z) \right) \times \left( \int\_{\mathbb{C}^d} |k\_\lambda(z)| \, dm(z) \right)^2 \\ &= \int\_{\mathbb{C}^d} |F(z)|^2 \sum\_{\lambda \in \Lambda} |k(z-\lambda)| \, dm(z) \times \int\_{\mathbb{C}^d} |k(z)| \, dm(z) . \end{split}$$

Here � **<sup>C</sup>***<sup>d</sup>* <sup>|</sup>*k*(*z*)<sup>|</sup> *dm*(*z*) = *<sup>m</sup>* <sup>&</sup>lt; <sup>∞</sup> because the kernel is integrable and also

$$\sum\_{\lambda \in \Lambda} |k(z - \lambda)| = \sum\_{\gamma \in (z - \Lambda)} |k(\gamma)|$$

is bounded independently of *<sup>z</sup>*, and since *<sup>z</sup>* <sup>−</sup> <sup>Λ</sup> has the same separation constant as <sup>Λ</sup> we can apply Lemma 4.8. Then

$$\begin{aligned} \sum\_{\lambda \in \Lambda} |F(\lambda)|^2 &\le \int\_{\mathbb{C}^d} |F(z)|^2 \sum\_{\gamma \in (z-\Lambda)} |k(\gamma)| \, dm(z) \int\_{\mathbb{C}^d} |k(z)| \, dm(z) \\ &\le \mathcal{B} \|F\|^2 \end{aligned}$$

where *<sup>B</sup>* <sup>=</sup> *<sup>m</sup> <sup>c</sup>ε*2*<sup>d</sup>* �*Mk*�<sup>1</sup> and *<sup>ε</sup>* is the separation constant of <sup>Λ</sup>.

Next, we want to know when the Gabor system *G*(*h*, Λ) is a frame for *L*2(**R***d*). First we observe that *Vh <sup>f</sup>*(*z*) = �*<sup>f</sup>* , *hz*� and �*Vh <sup>f</sup>* � <sup>=</sup> � *<sup>f</sup>* � (if �*h*� <sup>=</sup> 1). As we have bijective correspondence between <sup>G</sup>*<sup>h</sup>* and *<sup>L</sup>*2(**R***d*) by the Gabor transform, we can write

$$\sum\_{\lambda \in \Lambda} |F(\lambda)|^2 = \sum\_{\lambda \in \Lambda} |V\_{\hbar} f(\lambda)|^2 = \sum\_{\lambda \in \Lambda} |\langle f, h\_{\lambda} \rangle|^2$$

for every *<sup>f</sup>* <sup>∈</sup> *<sup>L</sup>*2(**R***d*) or for every *Vh <sup>f</sup>* ∈ G*h*. Therefore the frame condition and that of sampling set are equivalent. We conclude that: given a discrete set <sup>Λ</sup> <sup>⊂</sup> **<sup>C</sup>***<sup>d</sup>* and a Gabor window *<sup>h</sup>* <sup>∈</sup> *<sup>L</sup>*2(**R***d*), *<sup>G</sup>*(*h*, <sup>Λ</sup>) is a frame for *<sup>L</sup>*2(**R***d*) if and only if <sup>Λ</sup> is a sampling set for <sup>G</sup>*h*.

Let <sup>Σ</sup>*<sup>α</sup>* <sup>=</sup> {*<sup>h</sup>* <sup>∈</sup> *<sup>C</sup>*∞(**R**) <sup>∩</sup> *<sup>L</sup>*2(**R**) : *<sup>h</sup>*′ <sup>+</sup> *zh* <sup>∈</sup> *<sup>L</sup>*2(**R**) *and* �*h*′ <sup>+</sup> *zh*�<sup>2</sup> <sup>≦</sup> *<sup>α</sup>*�*h*�<sup>2</sup> *f or z*(*t*) = *t t* <sup>∈</sup> **<sup>R</sup>**}. It is clear that *C*<sup>∞</sup> *<sup>c</sup>* (**R**) <sup>⊆</sup> <sup>Σ</sup>*<sup>α</sup>* and so <sup>Σ</sup>*<sup>α</sup>* is a non empty set.

**Lemma 4.10** Let *<sup>h</sup>* <sup>∈</sup> <sup>Σ</sup>*<sup>α</sup>* and let {*λn*}*n*∈**<sup>N</sup>** and {*µn*}*n*∈**<sup>N</sup>** be sequences of scalars and suppose that there exist positive numbers *B* and *L* such that

$$\sum\_{n} |F(\lambda\_n)|^2 \le B ||F||^2 ; (F \in \mathcal{G}\_h).$$

and

$$|\mu\_n - \lambda\_n| \le L\_\prime (n = 1, 2, 3, \dotsb).$$

Then for every *<sup>F</sup>* ∈ G*<sup>h</sup>*

$$\sum\_{n} |F(\lambda\_n) - F(\mu\_n)|^2 \le B(e^{aL} - 1)^2 ||F||^2.$$

**Proof.** Let *<sup>F</sup>* be an element of <sup>G</sup>*h*. By expanding *<sup>F</sup>* in a Taylor series about *<sup>λ</sup>n*, we find that

$$F(\mu\_n) - F(\lambda\_n) = \sum\_{k=1}^{\infty} \frac{F^{(k)}(\lambda\_n)}{k!} (\mu\_n - \lambda\_n)^k (n = 1, 2, 3, \dotsb).$$

If *ρ* is an arbitrary positive number, then by multiplying and dividing the summand by *ρ<sup>k</sup>* we find also that

$$|F(\mu\_n) - F(\lambda\_n)|^2 \le \sum\_{k=1}^{\infty} \frac{|F^{(k)}(\lambda\_n)|^2}{\rho^{2k} k!} \sum\_{k=1}^{\infty} \frac{\rho^{2k} |\mu\_n - \lambda\_n|^{2k}}{k!}$$

Since <sup>G</sup>*<sup>h</sup>* is closed under differentiation and �*F*′ �<sup>2</sup> <sup>≦</sup> *<sup>α</sup>*�*F*�<sup>2</sup> it follows that

$$\begin{aligned} \sum\_{n} |F^{(k)}(\lambda\_n)|^2 &\le B \|F^{(k)}\|^2 \\ &\le B \alpha^{2k} \|F\|^2 \quad k = 1, 2, 3, \dots \end{aligned}$$

Therefore we obtain

$$\begin{aligned} \sum |F(\lambda\_n) - F(\mu\_n)|^2 &\le B \|F\|^2 \sum\_{k=1}^\infty \frac{\alpha^{2k}}{\rho^{2k} k!} \sum\_{k=1}^\infty \frac{(\rho L)^{2k}}{k!} \\ &= B \|F\|^2 (e^{\frac{\mu^2}{\rho^2}} - 1)(e^{\rho^2 L^2} - 1) \end{aligned}$$

10.5772/60034

since <sup>|</sup>*µ<sup>n</sup>* <sup>−</sup> *<sup>λ</sup>n*<sup>|</sup> <sup>≦</sup> *<sup>L</sup>*.

24 Fourier Transform

It is clear that *C*<sup>∞</sup>

Then for every *<sup>F</sup>* ∈ G*<sup>h</sup>*

we find also that

Therefore we obtain

and

for every *<sup>f</sup>* <sup>∈</sup> *<sup>L</sup>*2(**R***d*) or for every *Vh <sup>f</sup>* ∈ G*h*. Therefore the frame condition and that of sampling set are equivalent. We conclude that: given a discrete set <sup>Λ</sup> <sup>⊂</sup> **<sup>C</sup>***<sup>d</sup>* and a Gabor window *<sup>h</sup>* <sup>∈</sup> *<sup>L</sup>*2(**R***d*), *<sup>G</sup>*(*h*, <sup>Λ</sup>) is a frame for *<sup>L</sup>*2(**R***d*) if and only if <sup>Λ</sup> is a sampling set for <sup>G</sup>*h*. Let <sup>Σ</sup>*<sup>α</sup>* <sup>=</sup> {*<sup>h</sup>* <sup>∈</sup> *<sup>C</sup>*∞(**R**) <sup>∩</sup> *<sup>L</sup>*2(**R**) : *<sup>h</sup>*′ <sup>+</sup> *zh* <sup>∈</sup> *<sup>L</sup>*2(**R**) *and* �*h*′ <sup>+</sup> *zh*�<sup>2</sup> <sup>≦</sup> *<sup>α</sup>*�*h*�<sup>2</sup> *f or z*(*t*) = *t t* <sup>∈</sup> **<sup>R</sup>**}.

**Lemma 4.10** Let *<sup>h</sup>* <sup>∈</sup> <sup>Σ</sup>*<sup>α</sup>* and let {*λn*}*n*∈**<sup>N</sup>** and {*µn*}*n*∈**<sup>N</sup>** be sequences of scalars and suppose

<sup>|</sup>*µ<sup>n</sup>* <sup>−</sup> *<sup>λ</sup>n*<sup>|</sup> <sup>≦</sup> *<sup>L</sup>*,(*<sup>n</sup>* <sup>=</sup> 1, 2, 3, ···).

**Proof.** Let *<sup>F</sup>* be an element of <sup>G</sup>*h*. By expanding *<sup>F</sup>* in a Taylor series about *<sup>λ</sup>n*, we find that

If *ρ* is an arbitrary positive number, then by multiplying and dividing the summand by *ρ<sup>k</sup>*

<sup>|</sup>*F*(*k*)(*λn*)<sup>|</sup>

*ρ*<sup>2</sup>*kk*!

�2

<sup>2</sup> <sup>≦</sup> *<sup>B</sup>*�*F*�<sup>2</sup> <sup>∞</sup>

<sup>=</sup> *<sup>B</sup>*�*F*�2(*<sup>e</sup>*

2

<sup>≦</sup> *<sup>B</sup>α*2*k*�*F*�2, *<sup>k</sup>* <sup>=</sup> 1, 2, 3, ···

∑ *k*=1

> *α*2 *ρ*2

*α*2*k ρ*<sup>2</sup>*kk*!

<sup>−</sup> <sup>1</sup>)(*<sup>e</sup>*

∞ ∑ *k*=1

*ρ*2*L*<sup>2</sup>

(*ρL*)2*<sup>k</sup> k*!

<sup>−</sup> <sup>1</sup>)

∞ ∑ *k*=1 *<sup>ρ</sup>*2*k*|*µ<sup>n</sup>* <sup>−</sup> *<sup>λ</sup>n*<sup>|</sup>

�<sup>2</sup> <sup>≦</sup> *<sup>α</sup>*�*F*�<sup>2</sup> it follows that

*k*!

2*k*

*F*(*k*)(*λn*)

<sup>2</sup> <sup>≦</sup> *<sup>B</sup>*�*F*(*k*)

<sup>2</sup> ≦ *B*(*e*

*αL*

<sup>−</sup> <sup>1</sup>)2�*F*�2.

*<sup>k</sup>*! (*µ<sup>n</sup>* <sup>−</sup> *<sup>λ</sup>n*)*k*(*<sup>n</sup>* <sup>=</sup> 1, 2, 3, ···).

<sup>2</sup> <sup>≦</sup> *<sup>B</sup>*�*F*�2;(*<sup>F</sup>* ∈ G*h*)

*<sup>c</sup>* (**R**) <sup>⊆</sup> <sup>Σ</sup>*<sup>α</sup>* and so <sup>Σ</sup>*<sup>α</sup>* is a non empty set.

<sup>|</sup>*F*(*λn*)<sup>|</sup>

<sup>|</sup>*F*(*λn*) <sup>−</sup> *<sup>F</sup>*(*µn*)<sup>|</sup>

∞ ∑ *k*=1

<sup>2</sup> ≦ ∞ ∑ *k*=1

(*λn*)<sup>|</sup>

∑|*F*(*λn*) <sup>−</sup> *<sup>F</sup>*(*µn*)<sup>|</sup>

∑*n*

that there exist positive numbers *B* and *L* such that

∑*n*

*<sup>F</sup>*(*µn*) <sup>−</sup> *<sup>F</sup>*(*λn*) =

<sup>|</sup>*F*(*µn*) <sup>−</sup> *<sup>F</sup>*(*λn*)<sup>|</sup>

Since <sup>G</sup>*<sup>h</sup>* is closed under differentiation and �*F*′

∑*n* |*F*(*k*) Now by choosing *ρ* = *α <sup>L</sup>* we get

$$\sum\_{n} |F(\lambda\_n) - F(\mu\_n)|^2 \le B(e^{\alpha L} - 1)^2 ||F||^2.$$

**Theorem 4.11** If {*λn*}*n*∈**<sup>N</sup>** is a sampling set for <sup>G</sup>*h*(*<sup>h</sup>* <sup>∈</sup> <sup>Σ</sup>*α*) then there exists positive constant *<sup>L</sup>* such that if {*µn*}*n*∈**<sup>N</sup>** satisfies <sup>|</sup>*λ<sup>n</sup>* <sup>−</sup> *<sup>µ</sup>n*<sup>|</sup> <sup>≦</sup> *<sup>L</sup>* for all *<sup>n</sup>*, then {*µn*}*n*∈**<sup>N</sup>** is also sampling set.

**Proof.** Since {*λn*}*n*∈**<sup>N</sup>** is a sampling set for <sup>G</sup>*h*, then there exist positive constants *<sup>A</sup>* and *<sup>B</sup>* such that

$$A\|F\|^2 \stackrel{\textstyle \in}{\equiv} \sum\_{n} |F(\lambda\_n)|^2 \stackrel{\textstyle \in}{\equiv} B\|F\|^2$$

for every function *<sup>F</sup>* belonging to the Gabor space <sup>G</sup>*h*. Let {*µn*}*n*∈**<sup>N</sup>** be complex scalars for which <sup>|</sup>*λ<sup>n</sup>* <sup>−</sup> *<sup>µ</sup>n*<sup>|</sup> <sup>≦</sup> *<sup>L</sup>*(*<sup>n</sup>* <sup>=</sup> 1, 2, 3, ···). It is to be shown that if *<sup>L</sup>* is sufficiently small, then similar inequalities hold for the *µn*'s.

By virtue of the previous lemma, for every *<sup>F</sup>* ∈ G*h*,

$$\sum\_{n} |F(\lambda\_n) - F(\mu\_n)|^2 \le B(e^{\varkappa L} - 1)^2 ||F||^2$$

and since �*F*�<sup>2</sup> <sup>≦</sup> <sup>1</sup> *<sup>A</sup>* <sup>∑</sup>*<sup>n</sup>* <sup>|</sup>*F*(*λn*)<sup>|</sup> <sup>2</sup> to have

$$\sum\_{n} |F(\lambda\_n) - F(\mu\_n)|^2 \le \mathbb{C} \sum\_{n} |F(\lambda\_n)|^2.$$

where *<sup>C</sup>* <sup>=</sup> *<sup>B</sup> <sup>A</sup>* (*eα<sup>L</sup>* <sup>−</sup> <sup>1</sup>)2. Applying Minkowski's inequality, we find that

$$\left|\sqrt{\sum |F(\lambda\_n)|^2} - \sqrt{\sum |F(\mu\_n)|^2}\right| \le \sqrt{C \sum |F(\lambda\_n)|^2}$$

and hence

$$\sqrt{A}(1-\sqrt{\mathbb{C}})\|F\| \le \sqrt{\sum |F(\mu\_n)|^2} \le \sqrt{B}(1+\sqrt{\mathbb{C}})\|F\|.$$

for every *<sup>F</sup>*. Since *<sup>C</sup>* is less than 1 if *<sup>L</sup>* is sufficiently small, {*µn*}*n*∈**<sup>N</sup>** is a sampling set for <sup>G</sup>*h*.

**Problem 4.12** Let *<sup>h</sup>*, *<sup>k</sup>* <sup>∈</sup> *<sup>L</sup>*2(**R**). It would be desirable to show that there exists *<sup>ε</sup>* <sup>&</sup>gt; 0 such that if �*<sup>k</sup>* <sup>−</sup> *<sup>h</sup>*� <sup>&</sup>lt; *<sup>ε</sup>* and {*λn*} is a sampling set for <sup>G</sup>*<sup>h</sup>* then {*λn*} is a sampling set for <sup>G</sup>*k*. If one can show this and *<sup>h</sup>* <sup>∈</sup> <sup>Σ</sup>*α*, then for each *<sup>k</sup>* <sup>∈</sup> *<sup>L</sup>*2(**R**) with �*<sup>k</sup>* <sup>−</sup> *<sup>h</sup>*� <sup>&</sup>lt; *<sup>ε</sup>* the stability result of Theorem 4.11 holds for <sup>G</sup>*<sup>k</sup>* as well. Now since <sup>Σ</sup>*<sup>α</sup>* is norm dense in *<sup>L</sup>*2(**R**), one could conclude that Theorem 4.11 holds for each *<sup>h</sup>* <sup>∈</sup> *<sup>L</sup>*2(**R**). At present we are not able to prove that a "small" perturbation does not effect sampling set.

## **Acknowledgement**

This is part of the Ph.D. thesis of the first author in Tarbiat Modares University. She would like to thank Tarbiat Modares University and MSRT for moral and financial support. Part of this work appeared in a joint paper of the authors in Journal of Sciences, Islamic Republic of Iran.

## **Author details**

Nafya Hameed Mohammad<sup>1</sup> and Massoud Amini<sup>2</sup>

\*Address all correspondence to: nafya.mohammad@su.edu.iq; mamini@modares.ac.ir

1 Department of Mathematics, College of Education, Salahaddin University, Erbil, Iraq

2 Department of Mathematics, Faculty of Mathematical Sciences, Tarbiat Modares University, Tehran, Iran

## **References**


10.5772/60034


26 Fourier Transform

**Acknowledgement**

**Author details**

University, Tehran, Iran

**References**

1989.

Iran.

Iraq

for every *<sup>F</sup>*. Since *<sup>C</sup>* is less than 1 if *<sup>L</sup>* is sufficiently small, {*µn*}*n*∈**<sup>N</sup>** is a sampling set for <sup>G</sup>*h*.

**Problem 4.12** Let *<sup>h</sup>*, *<sup>k</sup>* <sup>∈</sup> *<sup>L</sup>*2(**R**). It would be desirable to show that there exists *<sup>ε</sup>* <sup>&</sup>gt; 0 such that if �*<sup>k</sup>* <sup>−</sup> *<sup>h</sup>*� <sup>&</sup>lt; *<sup>ε</sup>* and {*λn*} is a sampling set for <sup>G</sup>*<sup>h</sup>* then {*λn*} is a sampling set for <sup>G</sup>*k*. If one can show this and *<sup>h</sup>* <sup>∈</sup> <sup>Σ</sup>*α*, then for each *<sup>k</sup>* <sup>∈</sup> *<sup>L</sup>*2(**R**) with �*<sup>k</sup>* <sup>−</sup> *<sup>h</sup>*� <sup>&</sup>lt; *<sup>ε</sup>* the stability result of Theorem 4.11 holds for <sup>G</sup>*<sup>k</sup>* as well. Now since <sup>Σ</sup>*<sup>α</sup>* is norm dense in *<sup>L</sup>*2(**R**), one could conclude that Theorem 4.11 holds for each *<sup>h</sup>* <sup>∈</sup> *<sup>L</sup>*2(**R**). At present we are not able to prove

This is part of the Ph.D. thesis of the first author in Tarbiat Modares University. She would like to thank Tarbiat Modares University and MSRT for moral and financial support. Part of this work appeared in a joint paper of the authors in Journal of Sciences, Islamic Republic of

\*Address all correspondence to: nafya.mohammad@su.edu.iq; mamini@modares.ac.ir

[1] Gabor, D., Theory of Communication, *J. Inst. Elec. Eng.* 93 (1946) 429-457.

1 Department of Mathematics, College of Education, Salahaddin University, Erbil,

2 Department of Mathematics, Faculty of Mathematical Sciences, Tarbiat Modares

[2] Folland, G. B., *Harmonic Analysis in Phase Space*, Princeton University Press, Princeton,

[3] Feichtinger, H. G. and Gr*o*¨chenig, K., Banach spaces related to integrable group representations and their atomic decompositions. I., *J. Funct. Anal.* 86(2) (1989) 307-340.

[4] Liang, T. and Juan, G., Fast parallel algorithms for discrete Gabor expansion and

[5] Wexler, J. and Raz, S., Discrete Gabor expansions, *Signal Processing* 21 (1990) 207-220.

[6] Borichev, A., Gröchenig, K. and Lyubarskii, Yu., Frame constants of Gabor frames near

[7] Ascensi, G. and Bruna, J., Model space results for the Gabor and wavelet transforms,

transform based on multirate filtering, *Science China* 55(2) (2012) 293-300.

the critical density, *J. Math. Pures Appl.* 94 (2010) 170-182.

*IEEE Trans. Inform. Theory* 55(5) (2009) 2250-2259.

that a "small" perturbation does not effect sampling set.

Nafya Hameed Mohammad<sup>1</sup> and Massoud Amini<sup>2</sup>

