Fourier Transforms for Generalized Fredholm Equations

Juan Manuel Velazquez Arcos, Ricardo Teodoro Paez Hernandez, Alejandro Perez Ricardez and Jaime Granados Samaniego

## Abstract

In this chapter we take the conventional Fredholm integral equations as a guideline to define a broad class of equations we name generalized Fredholm equations with a larger scope of applications. We show first that these new kind of equations are really vector-integral equations with the same properties but with redefined and also enlarged elements in its structure replacing the old traditional concepts like in the case of the source or inhomogeneous term with the generalized source useful for describing the electromagnetic wave propagation. Then we can apply a Fourier transform to the new equations in order to obtain matrix equations to both types, inhomogeneous and homogeneous generalized Fredholm equations. Meanwhile, we discover new properties of the field we can describe with this new technology, that is, mean; we recognize that the old concept of nuclear resonances is present in the new equations and reinterpreted as the brake of the confinement of the electromagnetic field. It is important to say that some segments involving mathematical details of our present work were published somewhere by us, as part of independent researches with different specific goals, and we recall them as a tool to give a sound support of the Fourier transforms.

Keywords: Fredholm equations, electromagnetic resonances, electromagnetic confinement, evanescent waves, left-hand materials, Fourier transforms, vector-matrix equations

## 1. Introduction

There is a very broad class of problems on physics that requires a tool that not only serves to handle the mathematical problem related to the solution of some differential equation describing the behavior of a system but that gives us an alternative description of them from a distinct point of view in a manner that allows us to discover some hidden physical properties, that is, we need to generalize the application of the Fourier transform from the conventional task to achieve a set of algebraic equations to a complete alternative formulation in terms of the Fourier transform of the integral Fredholm equations [1–5, 13, 17]. Many of the problems we want to consider are those related with vector fields like the electromagnetic. For this situation we dedicate the present chapter first to the integral equation formulation of the electromagnetic traveling waves, and then, by the application of the Fourier transform, we obtain finally a matrix-vector formulation [9, 10, 12, 14, 18]. To this end we go from the conventional Fredholm equations to new vectorintegral equations we name generalized Fredholm equations proving that really they have the same properties of the conventional scalar Fredholm equations. In the meantime we discover that the new formulation brings a resonant behavior solution when some specific conditions are accomplished. The resonant behavior can be associated with the physical phenomenon of a brake of confinement of the so-called evanescent waves [6–8, 10–12, 19, 20] which leaves the region known as the nearfield zone and is strongly related to the condition we name a left-hand material condition of the propagation media. The name left-hand material conditions describes the fact that are related with a negative refraction index observed in artificial materials created by man and we have used for describe the propagation media property in which in some embedded region the electromagnetic waves are diffracted like in a left-hand material. We find in the first part of the present chapter a brief discussion about the relation between the inhomogeneous generalized Fredholm equations or GIFE [9, 10, 12, 18] and the homogeneous generalized Fredholm equations or GHFE. The GHFE are behind the presence of the resonant behavior, and we show how a sudden change in a little set of physical parameters related to propagation properties triggers the brake of the confinement of the evanescent waves. Then we incorporate to our description the plasma sandwich model or PSM and their own parameters in order to propose that the change in these last parameters changes drastically the wave propagation properties of media. It is important to advise that our procedures are applied to continuous systems and therefore are strictly original, and only the topics related to the funds of the PSM were taken from previous works that involved discrete systems.

or

Fourier Transforms for Generalized Fredholm Equations

DOI: http://dx.doi.org/10.5772/intechopen.85993

or GHFE.

tion defined by

transform associated with frequency

so that Eq. (7) becomes

We also have

27

where we have defined the function

That is, is the Fourier transform of

ð4Þ

ð5Þ

ð6Þ

ð7Þ

ð8Þ

ð9Þ

ð10Þ

This equation resembles inhomogeneous Fredholm's integral equation (IFE) but not as defined in scalar conventional form, and we will prove below that is strictly the case, so we call it generalized inhomogeneous Fredholm's integral equation or GIFE and the homogeneous version generalized homogeneous Fredholm's equation

Also, we have used summation convention over n and defined the kernel:

The signal can be written in terms of a well-behaved non-null func-

On the other hand, we can express the Green's function in terms of its Fourier

For convenience, we return to Eq. (2), which can be written as

## 2. Beginning of the generalized Fredholm equations

In this section we will build the generalized Fredholm equations mentioned in the introduction of this chapter. To this end, we suppose that both electric and magnetic fields have the linearity property, and for this reason we can relate their values represented with the symbol at different times and places and . Due to the mentioned linearity of the wave equation, we can write (bearing in mind that we can have more general conditions different to empty space)

$$F^{m}(\mathbf{r},t) = F^{m(\circ)}(\mathbf{r},t) + \int\_{\substack{n=1\\V}} \stackrel{\infty}{\int} \mathcal{G}^{mn(\circ)}(\mathbf{r},t;\mathbf{r}',t') U^{mn}\left(\mathbf{r}'\right) F^{n}\left(\mathbf{r}',t'\right) d\mathbf{t}' d\mathbf{V}' \tag{1}$$

Here

$$G^{\kappa\mathfrak{m}(\circ)}(\mathbf{r},t;\mathbf{r}',t') \tag{2}$$

is the free Green's function, and the complex dispersion coefficients are which contain the complete linear or nonlinear space-dependent interaction, but only time-independent ones are considered. By interchanging the volume and time differentials on integrands in Eq. (1), we obtain

$$A^{\ast m}(\mathbf{r},t) = A^{\ast m(\cdot)}(\mathbf{r},t) + \int\_{-\pi}^{\pi} \sum\_{\kappa=1}^{3} \int\_{V} G^{\ast m(\cdot)}(\mathbf{r},t;\mathbf{r}',t') U^{\ast m}(\mathbf{r}') F^{\ast}(\mathbf{r}',t') d\mathscr{V}' dt' \quad (3)$$

Fourier Transforms for Generalized Fredholm Equations DOI: http://dx.doi.org/10.5772/intechopen.85993

or

the Fourier transform, we obtain finally a matrix-vector formulation [9, 10, 12, 14, 18]. To this end we go from the conventional Fredholm equations to new vectorintegral equations we name generalized Fredholm equations proving that really they have the same properties of the conventional scalar Fredholm equations. In the meantime we discover that the new formulation brings a resonant behavior solution when some specific conditions are accomplished. The resonant behavior can be associated with the physical phenomenon of a brake of confinement of the so-called evanescent waves [6–8, 10–12, 19, 20] which leaves the region known as the nearfield zone and is strongly related to the condition we name a left-hand material condition of the propagation media. The name left-hand material conditions describes the fact that are related with a negative refraction index observed in artificial materials created by man and we have used for describe the propagation media property in which in some embedded region the electromagnetic waves are diffracted like in a left-hand material. We find in the first part of the present chapter a brief discussion about the relation between the inhomogeneous generalized Fredholm equations or GIFE [9, 10, 12, 18] and the homogeneous generalized Fredholm equations or GHFE. The GHFE are behind the presence of the resonant behavior, and we show how a sudden change in a little set of physical parameters related to propagation properties triggers the brake of the confinement of the evanescent waves. Then we incorporate to our description the plasma sandwich model or PSM and their own parameters in order to propose that the change in these last parameters changes drastically the wave propagation properties of media. It is important to advise that our procedures are applied to continuous systems and therefore are strictly original, and only the topics related to the funds of the PSM

Fourier Transforms - Century of Digitalization and Increasing Expectations

were taken from previous works that involved discrete systems.

In this section we will build the generalized Fredholm equations mentioned in the introduction of this chapter. To this end, we suppose that both electric and magnetic fields have the linearity property, and for this reason we can relate their values represented with the symbol at different times and places and . Due to the mentioned linearity of the wave equation, we can write (bearing in

mind that we can have more general conditions different to empty space)

Gmnð Þ<sup>∘</sup> r; t; r

is the free Green's function, and the complex dispersion coefficients are which contain the complete linear or nonlinear space-dependent interaction, but only time-independent ones are considered. By interchanging the volume and time

0 ; t <sup>0</sup> � �

Umn r <sup>0</sup> � �

F<sup>n</sup> r 0 ; t <sup>0</sup> � � dt0 dV<sup>0</sup>

(1)

ð2Þ

ð3Þ

∞ð

�∞

2. Beginning of the generalized Fredholm equations

ð

V ∑ 3 n¼1

differentials on integrands in Eq. (1), we obtain

<sup>F</sup><sup>m</sup>ð Þ¼ <sup>r</sup>; <sup>t</sup> <sup>F</sup><sup>m</sup>ð Þ<sup>∘</sup> ð Þþ <sup>r</sup>; <sup>t</sup>

Here

26

$$F'''(\mathbf{r},t) = F^{\sigma \mathbf{n}^{\circ} \circ}(\mathbf{r},t) + \bigcup\_{\sim \, \, \Gamma}^{\approx} K^{\prime \ast \mathbf{n}^{\circ} \circ}(\mathbf{r},t; \mathbf{r}^{\prime},t^{\prime}) F^{\prime \ast}(\mathbf{r}^{\prime},t^{\prime}) dV^{\prime} dt^{\prime} \qquad (4)$$

This equation resembles inhomogeneous Fredholm's integral equation (IFE) but not as defined in scalar conventional form, and we will prove below that is strictly the case, so we call it generalized inhomogeneous Fredholm's integral equation or GIFE and the homogeneous version generalized homogeneous Fredholm's equation or GHFE.

Also, we have used summation convention over n and defined the kernel:

$$K^{\mathfrak{nn}(\circ)}(\mathbf{r},t;\mathbf{r}',t') = G^{\mathfrak{nn}(\circ)}(\mathbf{r},t;\mathbf{r}',t')U^{\mathfrak{nn}}(\mathbf{r}')\tag{5}$$

The signal can be written in terms of a well-behaved non-null function defined by

$$F''(\mathbf{r}',t') = \begin{cases} \ 0 \ \acute{y} \ t \ \mathbf{t} \in (-\infty,0) \cup (T,\infty) \\\ \qquad Z''(\mathbf{r}',t') \ \acute{y} \ t \ \mathbf{t}' \in [0,T] \end{cases} \tag{6}$$

For convenience, we return to Eq. (2), which can be written as

$$Z^{\mathfrak{m}}(\mathbf{r},t) = Z^{\mathfrak{m}\langle \circ \rangle}(\mathbf{r},t) + \sum\_{\ast=1}^{\mathsf{N}} \int\_{0}^{T} G^{\mathfrak{m}\langle \circ \rangle}(\mathbf{r},t;\mathbf{r}',t')U^{\ast \ast \ast}(\mathbf{r}')Z^{\mathfrak{m}}(\mathbf{r}',t')dt'dV' \quad (7)$$

On the other hand, we can express the Green's function in terms of its Fourier transform associated with frequency

$$G^{\prime \ast \ast (\circ)}({\bf r}, t; {\bf r}^{\prime}, t^{\prime}) = \mathop{\rm l}^{\prime \ast}\_{\exists \pi} G^{\prime \ast \ast (\circ)}\_{\omega}({\bf r}; {\bf r}^{\prime}) e^{i \diamond ({\bf r} - {\bf r}^{\prime})} d\boldsymbol{d} \mathbf{o} \tag{8}$$

so that Eq. (7) becomes

$$Z'''(\mathbf{r},t) = Z^{m(\cdot)}(\mathbf{r},t) + \frac{1}{2\pi} \sum\_{n=1}^{3} \int U^{mn}(\mathbf{r}') \prod\_{\alpha}^{n} \mathcal{C}^{m\mathbf{r}(\boldsymbol{\epsilon})}(\mathbf{r};\mathbf{r}') \mathcal{g}^{\pi}(\mathbf{r}',\boldsymbol{\alpha}) d\boldsymbol{\epsilon} d\boldsymbol{\nu} \, V' \quad (9)$$

where we have defined the function

$$\mathbf{g'''(r',\alpha)} = \int\_0^r e^{\alpha \mathbf{w}^\*} Z^m(\mathbf{r'}, t') dt' \tag{10}$$

That is, is the Fourier transform of We also have

Fourier Transforms - Century of Digitalization and Increasing Expectations

$$Z'''(\mathbf{r},t) = \varinjlim\_{\omega \quad \omega}^{\sim} e^{\iota \alpha \prime} \mathbf{g}'''(\mathbf{r}, \alpha o) da \mathbf{o} \tag{11}$$

Eqs. (16) and (17) comprise the basic tools needed to describe the forward transmission of information but, as we will see in the next chapter, an incomplete description for time reversal. We can use Eq. (16) to get experimental data on the components of since the Fourier transforms of the original signals

we may consider Eqs. (16) and (17) as our starting point instead of assuming that

Nowadays, there is not any device capable to manipulate electromagnetic signals

in the easy way; we can manipulate sound waves mostly when we make a time reverse on them. Nevertheless, we have proposed in another work a recipe to handle this problem, so we are convinced that the treatment of the time reversal process that we now describe corresponds to a completely possible fact. Suppose that we have recorded a signal during a time T and now the reversed signal returns

4. The role of the Fourier transforms assisting time reverse

This Eq. (18) can be written in terms of the function as

We can express Eq. (19) in terms of the Fourier transform

And recalling the Fourier transform for <sup>Z</sup><sup>n</sup>ð Þ <sup>r</sup>; <sup>t</sup> , this can be written as

there is no signal for t < 0.

Fourier Transforms for Generalized Fredholm Equations

DOI: http://dx.doi.org/10.5772/intechopen.85993

to site r. Then we can write

in the form

29

are known, we can measure the arriving signals . In practice,

ð18Þ

ð19Þ

ð20Þ

ð21Þ

ð22Þ

Substituting in Eq. (9) and performing some algebra, we obtain

$$\mathbf{g}'''(\mathbf{r},\omega o) = \mathbf{g}^{\ast \ast (\ast)}(\mathbf{r},\omega o) + \sum\_{\kappa=1}^{\ast} \int\_{V} U^{\kappa \ast}(\mathbf{r}') G\_{\kappa}^{\ast \ast \ast (\ast)}(\mathbf{r};\mathbf{r}') \mathbf{g}^{\ast}(\mathbf{r}',\omega o) d\alpha dV' \quad (12)$$

Now we introduce a very useful and powerful notation we call vector-matrix form for Eq. (12) (vectors have another vectors as components, and also matrices have matrices as components):

$$\mathbf{g}^{\mathfrak{m}(\circ)}(\boldsymbol{\iota}\boldsymbol{\nu}) = \left[\mathbf{1} - \mathbf{K}^{(\circ)}(\boldsymbol{\iota}\boldsymbol{\nu})\right]\_{\mathfrak{n}}^{\mathfrak{m}} \mathbf{g}^{\mathfrak{n}}(\boldsymbol{\iota}\boldsymbol{\nu}) \tag{13}$$

(Einstein summation convention was used here) where

$$\mathbf{K}^{(\circ)}\left(\mathbf{r};\mathbf{r}';\boldsymbol{\omega}\right) \equiv U^{mn}(\mathbf{r})G\_{\boldsymbol{\omega}}^{mn(\circ)}\left(\mathbf{r};\mathbf{r}'\right) \tag{14}$$

and also define

$$\int\_{V} U^{mn} \left( \mathbf{r'} \right) G\_{\alpha}^{mn \left( \circ \right)} \left( \mathbf{r}; \mathbf{r'} \right) \mathbf{g}^{n} dV' \equiv \mathbf{K}^{mn \left( \circ \right)} (\boldsymbol{\alpha}) \mathbf{g}^{n} (\mathbf{r})$$

## 3. The vector-matrix forward equation

Eq. (13) can be inverted formally as

$$\mathbf{g''}(\boldsymbol{\alpha}) = \left[\mathbf{l} - \mathbf{K}^{(\circ)}(\boldsymbol{\alpha})\right]^{-1} \mathbf{l}\_{\boldsymbol{\alpha}}^{\star} \mathbf{g}^{\star(\circ)}(\boldsymbol{\alpha}) \tag{15}$$

By means of the development of this equation, we find the generalized Neumann series [12] and obtain the Fourier transform of complete Green's function . The result is [1]

$$\mathbf{g''} (\boldsymbol{\omega}) = \left[ \mathbf{1} + \mathbf{K} (\boldsymbol{\omega}) \right]\_{\boldsymbol{\omega}}^{\boldsymbol{\pi}} \mathbf{g'''}^{\boldsymbol{\omega}(\boldsymbol{\epsilon})} (\boldsymbol{\omega}) \tag{16}$$

Here we have defined

$$\mathbf{K}\_{n}^{m}\left(\mathbf{r};\mathbf{r}';o\right) = U^{mn}\left(\mathbf{r}'\right)G\_{o}^{mn}\left(\mathbf{r};\mathbf{r}'\right) \tag{17}$$

and the integral

$$\int\_{V} U^{mn}\left(\mathbf{r'}\right) G^{mn}\_{\alpha\nu}\left(\mathbf{r};\mathbf{r'}\right) \mathbf{g}^{m(\circ)}\left(\mathbf{r'}\right) dV' \equiv \mathbf{K}^{mn(\circ)}(\alpha) \mathbf{g}^{m(\circ)}(\mathbf{r})$$

Fourier Transforms for Generalized Fredholm Equations DOI: http://dx.doi.org/10.5772/intechopen.85993

ð11Þ

ð12Þ

ð13Þ

(14)

ð15Þ

ð16Þ

(17)

Substituting in Eq. (9) and performing some algebra, we obtain

Fourier Transforms - Century of Digitalization and Increasing Expectations

(Einstein summation convention was used here)

Kð Þ<sup>∘</sup> r; r 0 ;ω � �

> Gmnð Þ<sup>∘</sup> <sup>ω</sup> r; r <sup>0</sup> � �

have matrices as components):

where

and also define

ð

Umn r <sup>0</sup> � �

3. The vector-matrix forward equation

Eq. (13) can be inverted formally as

. The result is [1]

K<sup>m</sup> <sup>n</sup> r; r 0 ;ω � �

Here we have defined

ð

Umn r <sup>0</sup> � � Gmn <sup>ω</sup> r; r <sup>0</sup> � �

V

and the integral

28

V

Now we introduce a very useful and powerful notation we call vector-matrix form for Eq. (12) (vectors have another vectors as components, and also matrices

� <sup>U</sup>mnð Þ<sup>r</sup> <sup>G</sup>mnð Þ<sup>∘</sup> <sup>ω</sup> <sup>r</sup>; <sup>r</sup>

gndV<sup>0</sup>

By means of the development of this equation, we find the generalized Neumann series [12] and obtain the Fourier transform of complete Green's function

> <sup>¼</sup> <sup>U</sup>mn <sup>r</sup> <sup>0</sup> � � Gmn <sup>ω</sup> r; r <sup>0</sup> � �

g<sup>m</sup>ð Þ<sup>∘</sup> r <sup>0</sup> � � dV<sup>0</sup> <sup>0</sup> � �

� <sup>K</sup>mnð Þ<sup>∘</sup> ð Þ <sup>ω</sup> <sup>g</sup><sup>n</sup>ð Þ<sup>r</sup>

� <sup>K</sup>mnð Þ<sup>∘</sup> ð Þ <sup>ω</sup> <sup>g</sup><sup>m</sup>ð Þ<sup>∘</sup> ð Þ<sup>r</sup>

Eqs. (16) and (17) comprise the basic tools needed to describe the forward transmission of information but, as we will see in the next chapter, an incomplete description for time reversal. We can use Eq. (16) to get experimental data on the components of since the Fourier transforms of the original signals are known, we can measure the arriving signals . In practice, we may consider Eqs. (16) and (17) as our starting point instead of assuming that there is no signal for t < 0.

## 4. The role of the Fourier transforms assisting time reverse

Nowadays, there is not any device capable to manipulate electromagnetic signals in the easy way; we can manipulate sound waves mostly when we make a time reverse on them. Nevertheless, we have proposed in another work a recipe to handle this problem, so we are convinced that the treatment of the time reversal process that we now describe corresponds to a completely possible fact. Suppose that we have recorded a signal during a time T and now the reversed signal returns to site r. Then we can write

ð18Þ

This Eq. (18) can be written in terms of the function as

$$\begin{aligned} Z^n(\mathbf{r}, T - t) &= Z^{n(\circ)}(\mathbf{r}, T - t) \\ &+ \int\limits\_{V} \int U^{mn^\*}(\mathbf{r'}) G^{mn(\circ)\*}(\mathbf{r'}, T - t'; \mathbf{r}, t) \sum\_{m=1}^3 Z^m(\mathbf{r'}, T - t') dt' dV' \end{aligned} \tag{19}$$

We can express Eq. (19) in terms of the Fourier transform

$$G^{\text{soft}(\ast)^{\bullet}}(\mathbf{r}^{\prime}, T - t^{\prime}; \mathbf{r}, t) = \mathop{\frac{!}{2\pi}} \limits\_{-\infty}^{\infty} G^{\text{soft}(\ast)^{\bullet}}(\mathbf{r}^{\prime}, \mathbf{r}) e^{i\mathbf{n}\cdot(T - t^{\prime} - t^{\prime})} d\mathbf{n} \tag{20}$$

in the form

$$\begin{aligned} Z^n(\mathbf{r}, T - t) &= Z^{n(\circ)}(\mathbf{r}, T - t) \\ &+ \sum\_{m=1}^3 \iint \frac{1}{2\pi} U^{mn}(\mathbf{r}') \int\_{-\infty}^{\cdot} G\_{-\mathbf{q}}^{m(\circ)\dagger}(\mathbf{r}'; \mathbf{r}) e^{i\mathbf{q}\cdot(T - t' - t)} d\boldsymbol{\alpha} \mathcal{Z}^m(\mathbf{r}', T - t') dt' dV \end{aligned} \tag{21}$$

And recalling the Fourier transform for <sup>Z</sup><sup>n</sup>ð Þ <sup>r</sup>; <sup>t</sup> , this can be written as

$$\log''(\mathbf{r}, \mathfrak{o}) = \lg''^{\circ(\circ)}(\mathbf{r}, \mathfrak{o}) + \sum\_{\mathfrak{u}=\mathfrak{i}}^{\circ} \int\_{V} \lg''(\mathbf{r}', \mathfrak{o}) U^{\circ \circ \circ}(\mathbf{r}') G\_{\mathfrak{o}}^{\circ \circ \circ \circ \circ}(\mathbf{r}'; \mathbf{r}) e^{-i\mathfrak{u} \cdot \mathbf{r}} dV' \ (22)$$

At this point it is important to distinguish between functions related to forward phenomena and those related to backward direction when necessary. So we will use a different notation for both cases, and also, we introduce a quantum mechanics resembling notation for the product between matrices and vectors; in this manner, Eq. (22) can be written in vector (row vector) form like

or

31

Now we substitute in Eq. (29) this last expression for

Fourier Transforms for Generalized Fredholm Equations

DOI: http://dx.doi.org/10.5772/intechopen.85993

We then obtain the Neumann series [12] for the Fourier transform of the inte-

gral equation solution for time reversal (for reference see Eq. (18)):

Canceling parentheses we obtain

ð30Þ

ð31Þ

ð32Þ

ð33Þ

$$\mathcal{J}^{\ \mathfrak{n}(\cdot)}(\mathbf{r},\alpha) = \mathcal{J}^{\ \mathfrak{n}}(\mathbf{r},\alpha) - \left\langle \begin{array}{c} \mathcal{J}^{\ \mathfrak{n}}(\mathbf{r}',\alpha) \mathbf{M}^{\ \mathfrak{m}(\cdot)^{\flat}}(\mathbf{r};\mathbf{r}';\alpha) \end{array} \right\rangle \qquad (23)$$

where we introduced the quantum mechanics resembling notation:

$$\left\langle \int \mathcal{J}^{\ \nkern-1.2mu \text{ $\rm N \; \n\vdash n$ }}(\mathbf{r}',a\nu) \mathbf{M}^{\text{\tiny\rm m\&}{\rm r\&}{\rm r\&}}(\mathbf{r};\mathbf{r}';a\nu) \right\rangle = \int\_{V} \mathcal{J}^{\ \n\nvdash \rm r\&}(\mathbf{r}',a\nu) \mathbf{M}^{\text{\tiny\rm m\&}{\rm r\&}{\rm r\&}}(\mathbf{r};\mathbf{r}';a\nu) \,dV' \quad (24)$$

Also we define

$$\mathbf{M}^{\text{new}(\circ)\circ}(\mathbf{r}';\mathbf{r};\alpha) \equiv U^{\text{new}}(\mathbf{r}') \\ \mathbf{G}\_{\alpha}^{\text{new}(\circ)\circ}(\mathbf{r}';\mathbf{r}) \equiv U^{\text{new}}(\mathbf{r}') \\ G\_{\alpha}^{\text{new}(\circ)\circ}(\mathbf{r}';\mathbf{r}) \qquad (25)$$

and

$$\int\_{V} \mathcal{G} \, \prescript{\text{\tiny{\bf{v}}}}{}{\text{\tiny{\bf{v}}}} U^{\text{\tiny{\bf{v}}} \text{\tiny{\bf{m}}}}(\mathbf{r}') \mathcal{G}\_{\text{\tiny{\bf{a}}}} \mathcal{G}\_{\text{\tiny{\bf{m}}}} (\mathbf{r}'; \mathbf{r}) dV' \equiv \mathcal{G} \, \prescript{\text{\tiny{\bf{m}}}}{}{\text{\tiny{\bf{m}}}} (\mathbf{\mathcal{O}}) \tag{26}$$

Factorizing in Eq. (28) and using definition Eq. (29)

$$\mathcal{J}\_{\bullet} \triangleq \left. \mathcal{J}^{\prime \prime \prime} (\mathbf{r}, \boldsymbol{\alpha}) \right| \left. \mathbb{I} - \mathbf{M}^{\prime \prime \prime} (\mathbf{r}^{\prime}; \mathbf{r}; \boldsymbol{\alpha}) \right|\_{\ast}^{\ast} \equiv \left\langle \begin{array}{c} \mathcal{J}^{\prime \prime} (\mathbf{r}^{\prime}; \boldsymbol{\alpha}) \|\, \mathbb{I} \delta (\mathbf{r}^{\prime} - \mathbf{r}) - \mathbf{M}^{\prime \prime \prime} (\mathbf{r}^{\prime}; \mathbf{r}; \boldsymbol{\alpha}) \right\|\_{\ast}^{\ast} \right\rangle \tag{27}$$

In the following we will use systematically Eqs. (23), (25), and (27).

## 5. Fourier transforms and Neumann series make up a powerful tool

It is possible to invert formally Eq. (27)

$$\mathcal{J}^{\nkern-1.2ex}(\mathbf{r},\boldsymbol{\alpha}) \equiv \mathcal{J}^{\nkern-1.2ex}(\mathbf{r}',\boldsymbol{\alpha}) \big[ \mathbf{1} - \mathbf{M}^{\left(\ast\right)^{\nu}}(\mathbf{r}';\mathbf{r};\boldsymbol{\alpha}) \big]^{-1} \big[ \prescript{\boldsymbol{\pi}}{} \tag{28}$$

Formally

$$\mathbf{g}^{\boldsymbol{\alpha}}(\mathbf{r},\boldsymbol{\alpha}) = \sum\_{m=1}^{3} \left[ \boldsymbol{\delta}\_{m}^{\boldsymbol{\alpha}} \mathbf{g}^{m \boldsymbol{\alpha}}(\mathbf{r},\boldsymbol{\alpha}) \right]$$

$$\mathbf{g}^{\boldsymbol{\alpha}^{\boldsymbol{m}(\boldsymbol{\alpha})}}(\mathbf{r}',\boldsymbol{\alpha}) \mathbf{M}^{\boldsymbol{m}(\boldsymbol{\alpha})^{\boldsymbol{\alpha}}}(\mathbf{r}';\mathbf{r};\boldsymbol{\alpha}) + \boldsymbol{\mathcal{g}}^{\boldsymbol{m}(\boldsymbol{\alpha})}(\mathbf{r}',\boldsymbol{\alpha}) \left[ \mathbf{M}^{\boldsymbol{m}(\boldsymbol{\alpha})^{\boldsymbol{\alpha}}}(\mathbf{r}';\mathbf{r};\boldsymbol{\alpha}) \right]^{2}$$

$$+ \boldsymbol{\mathcal{g}}^{\boldsymbol{m}(\boldsymbol{\alpha})}(\mathbf{r}',\boldsymbol{\alpha}) \left[ \mathbf{M}^{\boldsymbol{m}(\boldsymbol{\alpha})^{\boldsymbol{\alpha}}}(\mathbf{r}';\mathbf{r};\boldsymbol{\alpha}) \right]^{3} + \cdots \tag{29}$$

Fourier Transforms for Generalized Fredholm Equations DOI: http://dx.doi.org/10.5772/intechopen.85993

or

At this point it is important to distinguish between functions related to forward phenomena and those related to backward direction when necessary. So we will use a different notation for both cases, and also, we introduce a quantum mechanics resembling notation for the product between matrices and vectors; in this manner,

where we introduced the quantum mechanics resembling notation:

ð23Þ

ð24Þ

ð25Þ

ð26Þ

ð27Þ

ð28Þ

ð29Þ

Eq. (22) can be written in vector (row vector) form like

Fourier Transforms - Century of Digitalization and Increasing Expectations

Factorizing in Eq. (28) and using definition Eq. (29)

It is possible to invert formally Eq. (27)

In the following we will use systematically Eqs. (23), (25), and (27).

5. Fourier transforms and Neumann series make up a powerful tool

Also we define

and

Formally

30

$$\begin{aligned} \mathbf{g}^{\boldsymbol{\alpha}}(\mathbf{r},\boldsymbol{\alpha}) &= \sum\_{n=1}^{\mathcal{I}} \left( \boldsymbol{\delta}\_{\boldsymbol{w}}^{\boldsymbol{\alpha}} \mathbf{g}^{\boldsymbol{\alpha}\prime \boldsymbol{\epsilon}}(\mathbf{r},\boldsymbol{\alpha}) \right) + \int\_{\mathcal{V}} \mathbf{g}^{\boldsymbol{w}\prime \boldsymbol{\epsilon}}(\mathbf{r}\prime,\boldsymbol{\alpha}) U^{\boldsymbol{w}\prime \boldsymbol{\epsilon}}(\mathbf{r}\prime) G\_{\boldsymbol{\alpha}}^{\boldsymbol{\alpha}\prime \boldsymbol{\epsilon}\prime}(\mathbf{r}\prime,\mathbf{r}) e^{-i\boldsymbol{\alpha}\prime \boldsymbol{\epsilon}} dV \,\boldsymbol{\epsilon} \\\\ &+ \int\_{\mathcal{V}} \int\_{\mathcal{V}} \mathbf{g}^{\boldsymbol{w}\prime \boldsymbol{\epsilon}}(\mathbf{r}\prime,\boldsymbol{\alpha}) U^{\boldsymbol{w}\prime \boldsymbol{\epsilon}}(\mathbf{r}\prime) G\_{\boldsymbol{\alpha}}^{\boldsymbol{w}\prime \boldsymbol{\epsilon}\prime}(\mathbf{r}\prime) e^{-i\boldsymbol{\alpha}\prime \boldsymbol{\epsilon}} U^{\boldsymbol{w}\prime \boldsymbol{\epsilon}}(\mathbf{r}\prime) G\_{\boldsymbol{\alpha}}^{\boldsymbol{w}\prime \boldsymbol{\epsilon}\prime}(\mathbf{r}\prime,\mathbf{r}\prime) e^{-i\boldsymbol{\alpha}\prime \boldsymbol{\epsilon}} dV \,\boldsymbol{\epsilon} \,\boldsymbol{\epsilon} \,\boldsymbol{\epsilon} \,\boldsymbol{\epsilon} \,\boldsymbol{\epsilon} \end{aligned} (30)$$

Now we substitute in Eq. (29) this last expression for

$$\mathbf{g}^{n}(\mathbf{r},\alpha\mathbf{o}) = \sum\_{a=1}^{\uparrow} \quad \left[\boldsymbol{\delta}\_{a}^{\prime}\mathbf{g}^{m(\cdot)}(\mathbf{r},\alpha\mathbf{o})\right]$$

$$+ \int\_{V} \boldsymbol{U}^{mn^{\*}}(\mathbf{r}^{\prime})\boldsymbol{G}\_{a0}^{m(\cdot)\prime}(\mathbf{r}^{\prime};\mathbf{r})e^{-i\alpha\mathbf{r}}\left\{\boldsymbol{g}^{m(\cdot)}(\mathbf{r}^{\prime},\alpha\mathbf{o})dV^{\*}\right.$$

$$+ \int\_{V} \boldsymbol{U}^{mn^{\*}}(\mathbf{r}^{\prime})\boldsymbol{G}\_{a}^{m(\cdot)\prime}(\mathbf{r}^{\prime};\mathbf{r})e^{-i\alpha\mathbf{r}}\boldsymbol{U}^{mn^{\*}}(\mathbf{r}^{\prime})\boldsymbol{G}\_{a}^{m(\cdot)\prime}(\mathbf{r}^{\prime};\mathbf{r}^{\prime})e^{-i\alpha\mathbf{r}}\boldsymbol{g}^{m(\cdot)}(\mathbf{r}^{\prime},\alpha\mathbf{o})dV^{\*}dV^{\prime} + \cdots \quad\left[\right] \tag{31}$$

Canceling parentheses we obtain

$$\mathbf{g}^{\boldsymbol{\alpha}}(\mathbf{r},\boldsymbol{\alpha}\boldsymbol{\alpha}) = \sum\_{m=1}^{\mathcal{I}} \int \mathbb{G}\_{m}^{\boldsymbol{\alpha}} \mathbf{g}^{m(\boldsymbol{\cdot})}(\mathbf{r},\boldsymbol{\alpha}\boldsymbol{\alpha})$$

$$+ \int \limits\_{\boldsymbol{\mathcal{V}}} U^{mn^{\boldsymbol{\alpha}}}(\mathbf{r}') G\_{\boldsymbol{\alpha}}^{\boldsymbol{\alpha}\mathbf{m}(\boldsymbol{\cdot})\mathbf{r}}(\mathbf{r}';\mathbf{r}) e^{-i\boldsymbol{\alpha}\boldsymbol{\mathsf{T}}} \left\{ \mathbf{g}^{m(\boldsymbol{\cdot})}(\mathbf{r}',\boldsymbol{\alpha}\boldsymbol{\alpha}) dV' \right.$$

$$+ \left[ \int\_{\mathcal{V}} U^{mn^{\boldsymbol{\alpha}}}(\mathbf{r}') G\_{\boldsymbol{\alpha}}^{m(\boldsymbol{\cdot})}(\mathbf{r}';\mathbf{r}) e^{-i\boldsymbol{\alpha}\boldsymbol{\mathsf{T}}} U^{mn^{\boldsymbol{\alpha}}}(\mathbf{r}'') G\_{\boldsymbol{\alpha}}^{m(\boldsymbol{\cdot})}(\mathbf{r}'';\mathbf{r}) e^{-i\boldsymbol{\alpha}\boldsymbol{\mathsf{T}}} g^{m(\boldsymbol{\cdot})}(\mathbf{r}'',\boldsymbol{\alpha}\boldsymbol{\mathsf{o}}) dV'' + \cdots \right] \qquad (32)$$

We then obtain the Neumann series [12] for the Fourier transform of the integral equation solution for time reversal (for reference see Eq. (18)):

$$\begin{aligned} \mathbf{g}''(\mathbf{r},\boldsymbol{\alpha}) &= \sum\_{\sigma=1}^{3} \int \boldsymbol{\delta}\_{\sigma}'' \boldsymbol{\varepsilon}^{m(\boldsymbol{\gamma})}(\mathbf{r},\boldsymbol{\alpha}) \\ &+ \int\_{\boldsymbol{V}} U^{m\boldsymbol{\gamma}\prime}(\mathbf{r}') G\_{\boldsymbol{\alpha}}^{m(\boldsymbol{\gamma})\prime}(\mathbf{r}\prime;\mathbf{r}) e^{-i\boldsymbol{\alpha}\prime T} \\ \\ &+ \int\_{\boldsymbol{V}} G\_{\boldsymbol{\alpha}}^{m(\boldsymbol{\gamma})\prime}(\mathbf{r}\prime;\mathbf{r}\prime) e^{-i\boldsymbol{\alpha}\prime} U^{m\boldsymbol{\gamma}\prime}(\mathbf{r}\prime) G\_{\boldsymbol{\alpha}}^{m(\boldsymbol{\gamma})\prime}(\mathbf{r}\prime;\mathbf{r}) e^{-i\boldsymbol{\alpha}\prime} dV \,\boldsymbol{\gamma} \\ \\ &+ \int\_{\boldsymbol{V}} G\_{\boldsymbol{\alpha}}^{m(\boldsymbol{\gamma})\prime}(\mathbf{r}\prime;\mathbf{r}\prime) e^{-i\boldsymbol{\alpha}\prime} U^{m\boldsymbol{\gamma}\prime}(\mathbf{r}\prime) G\_{\boldsymbol{\alpha}}^{m(\boldsymbol{\gamma})\prime}(\mathbf{r}\prime;\mathbf{r}\prime) e^{-i\boldsymbol{\alpha}\prime} dV \,\boldsymbol{\gamma} \,\boldsymbol{\delta} \,\boldsymbol{\gamma} \,\mathrm{d}V \,\frac{1}{2} \mathrm{d}V \,\frac{1}{2} \mathrm{d}V \end{aligned} (33)$$

## 6. An algebraic equation for time reverse

Because the bracketed expression in Eq. (36) is convergent, then it must equal the Fourier transform of complete Green's function , so that we can write

$$\mathcal{G}\_{\mathcal{J}}\,^{\n\prime}(\mathbf{r},\mathcal{O}) = \sum\_{\nu=1}^{3} [\mathcal{S}\_{\nu}^{\prime}\mathcal{G}\_{\nu}\,^{\ast\prime\prime}(\mathbf{r},\mathcal{O})] + \int\_{\mathcal{V}} U^{\ast\ast\prime}(\mathbf{r}\prime)\mathcal{G}\_{\nu}^{\prime\ast\ast}(\mathbf{r}\prime;\mathbf{r})\,\_{\mathcal{J}}\mathcal{F}^{\prime\ast\prime}(\mathbf{r}\prime,\mathcal{O}) \,dV\,\,\mathcal{V} \,\tag{34}$$

Equation (34) can be written in a compact row vector form:

$$\mathcal{J}^{\ \prime\prime}(\boldsymbol{\alpha}) = \mathcal{J}^{\ \prime\prime\prime}(\boldsymbol{\alpha})[1 + \mathbf{M}^\*(\boldsymbol{\alpha})]\_{\boldsymbol{\kappa}}^{\prime} \tag{35}$$

ð41Þ

ð42Þ

ð43Þ

ð44Þ

ð45Þ

ð46Þ

ð47Þ

ð48Þ

ð51Þ

<sup>m</sup> (49)

Then using Eq. (16), we obtain

DOI: http://dx.doi.org/10.5772/intechopen.85993

Fourier Transforms for Generalized Fredholm Equations

are or as we have seen Now, by spanning Eq. (41)

This can be expressed as

Here we have defined the product

33

Then we can write

In this expression is also a short notation for a "vector" whose components

and by rearrangement of terms and writing only the operators

<sup>U</sup>nmð Þ<sup>r</sup> <sup>G</sup>mn

But we can now explicitly write Eq. (45) in terms of Green's function:

<sup>ω</sup> r; r

That is, the Fourier transform of Green's function satisfies the equation

And if we start with Eq. (39) (time reversal), we obtain by a similar procedure

<sup>0</sup> � ½ � UGð Þ <sup>ω</sup> <sup>n</sup>

<sup>G</sup>ð Þ¼ <sup>ω</sup> <sup>G</sup>ð Þ<sup>∘</sup> ð Þþ <sup>ω</sup> <sup>K</sup>ð Þ <sup>ω</sup> <sup>G</sup>ð Þ<sup>∘</sup> ð Þ <sup>ω</sup> (50)

or

In this equation, we define the kernel

$$\mathbf{M}^\*(\mathbf{r}';\mathbf{r};\boldsymbol{\phi}) \equiv U^{\star n \ast \ast}(\mathbf{r}') \mathcal{G}^{\star n \ast}\_{\iota \theta}(\mathbf{r}';\mathbf{r}) \tag{36}$$

and also define

$$\int\_{Y} \mathcal{G}^{\ \prime (\circ)}(\mathbf{r}') U^{\pi a^{\ast}}(\mathbf{r}') \mathcal{G}^{\pi \ast \prime}\_{\
u}(\mathbf{r}';\mathbf{r}) dV^{\ast} \equiv \mathcal{G}^{\ \prime (\circ)} \mathbf{M}^{\pi \ast \ast}(\boldsymbol{\mathcal{O}}) \tag{37}$$

Transposing Eq. (35) we obtain finally the column vector form (for real interactions):

$$\mathbf{g''} (\boldsymbol{\omega}) = [1 + \mathbf{M}(\boldsymbol{\omega})]\_{\boldsymbol{\omega}}^{\boldsymbol{\omega}} \mathbf{g'''}^{\boldsymbol{\omega}(\circ)}(\boldsymbol{\omega}) \tag{38}$$

Obviously, Eq. (38) is identical with Eq. (16) but with Mð Þ ω instead of Kð Þ ω .

## 7. Operators and resonances on continuum formulation

Eqs. (16) and (38) are algebraic representations of integral equations, that is, they are strongly dependent on the Fourier transform of the Green function; indeed the behavior of the late referred function determines the solution whether or not the regime was resonant. For this reason it is convenient to analyze how the Green function changes in the neighborhood of a resonance. With this purpose in mind, we recall Eqs. (13) and (16):

$$\mathbf{g}^{\mathfrak{m}(\circ)}(\boldsymbol{\omega}) = \left[\mathbf{1} - \mathbf{K}^{(\circ)}(\boldsymbol{\omega})\right]\_{\mathfrak{n}}^{\mathfrak{m}} \mathbf{g}^{\mathfrak{n}}(\boldsymbol{\omega}) \tag{39}$$

$$\mathbf{g}''(\boldsymbol{\omega}) = \left[\mathbf{1} + \mathbf{K}(\boldsymbol{\omega})\right]\_{\boldsymbol{\omega}}^{\boldsymbol{\pi}} \mathbf{g}^{\boldsymbol{\pi}(\boldsymbol{\epsilon})}(\boldsymbol{\omega}) \tag{40}$$

By applying the operator from the left to Eq. (13) and summing over m, we have

Fourier Transforms for Generalized Fredholm Equations DOI: http://dx.doi.org/10.5772/intechopen.85993

$$\left[\mathbf{1} + \mathbf{K}(\boldsymbol{\phi})\right]\_{n}^{\mathrm{v}} \mathbf{g}^{\mathrm{v}(\boldsymbol{\cdot})}(\boldsymbol{\omega}) = \left[\mathbf{1} + \mathbf{K}(\boldsymbol{\omega})\right]\_{n}^{\mathrm{v}} \left[\mathbf{1} - \mathbf{K}^{(\boldsymbol{\cdot})}(\boldsymbol{\omega})\right]\_{n}^{\mathrm{v}} \mathbf{g}^{\mathrm{v}}(\boldsymbol{\omega})\tag{41}$$

Then using Eq. (16), we obtain

$$\mathbf{g}^{\circ}(\boldsymbol{\omega}) = \left[\mathbf{1} + \mathbf{K}(\boldsymbol{\omega})\right]\_{\boldsymbol{\omega}}^{\circ} \left[\mathbf{1} - \mathbf{K}^{(\circ)}(\boldsymbol{\omega})\right]\_{\boldsymbol{\omega}}^{\circ} \mathbf{g}^{\circ}(\boldsymbol{\omega})\tag{42}$$

or

6. An algebraic equation for time reverse

In this equation, we define the kernel

and also define

we recall Eqs. (13) and (16):

m, we have

32

tions):

write

Because the bracketed expression in Eq. (36) is convergent, then it must equal the Fourier transform of complete Green's function , so that we can

Transposing Eq. (35) we obtain finally the column vector form (for real interac-

Obviously, Eq. (38) is identical with Eq. (16) but with Mð Þ ω instead of Kð Þ ω .

Eqs. (16) and (38) are algebraic representations of integral equations, that is, they are strongly dependent on the Fourier transform of the Green function; indeed the behavior of the late referred function determines the solution whether or not the regime was resonant. For this reason it is convenient to analyze how the Green function changes in the neighborhood of a resonance. With this purpose in mind,

By applying the operator from the left to Eq. (13) and summing over

7. Operators and resonances on continuum formulation

Equation (34) can be written in a compact row vector form:

Fourier Transforms - Century of Digitalization and Increasing Expectations

ð34Þ

ð35Þ

ð36Þ

ð37Þ

ð38Þ

ð39Þ

ð40Þ

$$\mathbf{g}(\boldsymbol{\alpha}) = \left[\mathbf{1} + \mathbf{K}(\boldsymbol{\alpha})\right] \left[\mathbf{1} - \mathbf{K}^{(\circ)}(\boldsymbol{\alpha})\right] \mathbf{g}(\boldsymbol{\alpha}) \tag{43}$$

In this expression is also a short notation for a "vector" whose components are or as we have seen

Now, by spanning Eq. (41)

$$\mathbf{g}(\boldsymbol{\alpha}) = \left[\mathbf{1} - \mathbf{K}^{(\circ)}(\boldsymbol{\alpha}) + \mathbf{K}(\boldsymbol{\alpha}) - \mathbf{K}(\boldsymbol{\alpha})\mathbf{K}^{(\circ)}(\boldsymbol{\alpha})\right] \mathbf{g}(\boldsymbol{\alpha})\tag{44}$$

This can be expressed as

$$\mathbf{g}(\boldsymbol{\varrho}) = \mathbf{1}\mathbf{g}(\boldsymbol{\varrho}) + \left[ -\mathbf{K}^{(\circ)}(\boldsymbol{\varrho}) + \mathbf{K}(\boldsymbol{\varrho}) - \mathbf{K}(\boldsymbol{\varrho})\mathbf{K}^{(\circ)}(\boldsymbol{\varrho}) \right] \mathbf{g}(\boldsymbol{\varrho}) \tag{45}$$

Then we can write

$$\left[-\mathbf{K}^{(\circ)}(\boldsymbol{\alpha}) + \mathbf{K}(\boldsymbol{\alpha}) - \mathbf{K}(\boldsymbol{\alpha})\mathbf{K}^{(\circ)}(\boldsymbol{\alpha})\right] \mathbf{g}(\boldsymbol{\alpha}) = 0\tag{46}$$

and by rearrangement of terms and writing only the operators

$$\mathbf{K}(\boldsymbol{\phi}) = \mathbf{K}^{(\circ)}(\boldsymbol{\phi}) + \mathbf{K}(\boldsymbol{\phi})\mathbf{K}^{(\circ)}(\boldsymbol{\phi})\tag{47}$$

But we can now explicitly write Eq. (45) in terms of Green's function:

$$\mathbf{G}(\boldsymbol{\alpha})\mathbf{U} = \mathbf{G}^{(\circ)}(\boldsymbol{\alpha})\mathbf{U} + \mathbf{G}(\boldsymbol{\alpha})\mathbf{U}\mathbf{G}^{(\circ)}(\boldsymbol{\alpha})\mathbf{U} \tag{48}$$

Here we have defined the product

$$\left[U^{nm}(\mathbf{r})G^{mn}\_{\boldsymbol{\alpha}}\left(\mathbf{r};\mathbf{r}'\right)\equiv\left[\mathbf{U}\mathbf{G}(\boldsymbol{\alpha})\right]^{n}\_{m}\tag{49}$$

That is, the Fourier transform of Green's function satisfies the equation

$$\mathbf{G}(\boldsymbol{\alpha}) = \mathbf{G}^{(\circ)}(\boldsymbol{\alpha}) + \mathbf{K}(\boldsymbol{\alpha})\mathbf{G}^{(\circ)}(\boldsymbol{\alpha})\tag{50}$$

And if we start with Eq. (39) (time reversal), we obtain by a similar procedure

$$\mathbf{G}(\alpha) = \mathbf{G}^{\{\circ\}}(\alpha) + \mathbf{M}(\alpha)\mathbf{G}^{\{\circ\}}(\alpha) \tag{51}$$

Now, if we are near a resonance, Eqs. (48) and (49) are transformed in homogeneous equations with solutions we will denote as weð Þ ω , and if we denote the interaction as <sup>U</sup> and the kernel <sup>K</sup>ð Þ<sup>∘</sup> ð Þ <sup>ω</sup> , then from Eqs. (48) or (49) without the source term, we have the following relation:

$$\mathbf{w}\_l^\dagger(\boldsymbol{\alpha})\mathbf{U}\mathbf{w}\_u(\boldsymbol{\alpha})\left[\boldsymbol{\eta}\_u^{-1} - \boldsymbol{\eta}\_l^{-1}\right] = \mathbf{0} \tag{52}$$

with also

Figure 1.

35

composed of unmagnetized plasma.

We can reduce the last equations to a compact one:

Fourier Transforms for Generalized Fredholm Equations

DOI: http://dx.doi.org/10.5772/intechopen.85993

It is clear that our procedure leads to an inhomogeneous Fredholm equation in which it is possible to observe that the transit from a non-resonant regime to a

In precedent sections we have seen how we can go from inhomogeneous to homogeneous Fredholm equations, that is, from non-resonant or conventional

We show the superposition of three plasma layers subjected to local high electromagnetic potential creating resonances and releasing evanescent waves: Layer M is composed of magnetic plasma, and the U layers are

resonant regime is described by the generalized source term <sup>Φ</sup>ð Þ<sup>∘</sup> ð Þ <sup>r</sup>;<sup>ω</sup> .

9. The role of resonances on broadcasting applications

ð57Þ

ð58Þ

This relation establishes that the resonant solutions are mutually orthogonal and the functions η ωð Þ are known as the Fredholm eigenvalues.

## 8. The homogeneous Fredholm equation and Fredholm's eigenvalue

As we saw in Section 8, the resonant solutions are orthogonal and in Eq. (50) the Fredholm eigenvalues appear, but these last functions emerge when the inhomogeneous Fredholm equations are transformed in a homogeneous equation near a resonance. The resulting homogeneous equation is

$$\boldsymbol{\eta}\boldsymbol{\eta}^{m}(\mathbf{r};\boldsymbol{\alpha}) = \boldsymbol{\eta}\_{e}(\boldsymbol{\alpha}) \stackrel{\sim}{\underset{\mathbf{0}}{\text{ }}} \mathbf{K}\_{n}^{m(\circ)}(\boldsymbol{\alpha};\mathbf{r},\mathbf{r}')\boldsymbol{\nu}\_{e}^{m}(\mathbf{r}';\boldsymbol{\alpha})dr' \tag{53}$$

According to the theory of homogeneous Fredholm equations [1, 2, 3, 5, 9, 15, 16], one of the conditions for the existence of solutions is that first Fredholm's minor complies

$$
\mathcal{M} \stackrel{\text{'''}}{=}(\mathbf{r}, \mathbf{r}\_0; \boldsymbol{\alpha}) = \eta(\boldsymbol{\alpha})\Delta(\boldsymbol{\eta}, \boldsymbol{\alpha})
$$

$$
+\eta(\boldsymbol{\alpha})\stackrel{\text{'''}}{\int}\_{\boldsymbol{\eta}} \mathbf{K}\_{\boldsymbol{\eta}}^{\text{\*\*}(\circ)}(\boldsymbol{\alpha}; \mathbf{r}, \mathbf{s})\mathcal{M} \stackrel{\text{''}}{=}(\mathbf{s}, \mathbf{r}\_0; \boldsymbol{\alpha})ds\tag{54}
$$

From Eqs. (51) and (52) and after a little algebra, we arrive to the following equation:

$$
\mathcal{M} \stackrel{\text{u}}{=} (\mathbf{r}, \mathbf{r}\_0; \boldsymbol{\alpha}) - \Delta(\boldsymbol{\eta}, \boldsymbol{\alpha}) \mathbf{g}^n(\mathbf{r}, \boldsymbol{\alpha}) = \\
$$

$$
\Delta(\boldsymbol{\eta}, \boldsymbol{\alpha}) [\boldsymbol{\eta}(\boldsymbol{\alpha}) - \mathbf{g}^{\boldsymbol{m}(\boldsymbol{\epsilon})}(\mathbf{r}, \boldsymbol{\alpha})]
$$

$$
+ \eta \bigcap\_{\boldsymbol{0}}^{\circ} \mathbf{K}\_{\boldsymbol{n}}^{\boldsymbol{m}(\boldsymbol{\epsilon})}(\boldsymbol{\alpha}; \mathbf{r}, \mathbf{s}) \Big[\mathcal{M} \stackrel{\text{u}}{=} (\mathbf{s}, \mathbf{r}\_0; \boldsymbol{\alpha}) - \Delta(\boldsymbol{\eta}, \boldsymbol{\alpha}) \mathbf{g}^n(\mathbf{r}', \boldsymbol{\alpha}) \Big] ds
$$

$$
+ \Delta(\boldsymbol{\eta}, \boldsymbol{\alpha}) [\boldsymbol{\eta}(\boldsymbol{\alpha}) - \boldsymbol{\upsilon}(\boldsymbol{\alpha})] \Big[\prod\_{\boldsymbol{0}}^{\circ} \mathbf{K}\_{\boldsymbol{n}}^{\boldsymbol{m}(\boldsymbol{\epsilon})}(\boldsymbol{\alpha}; \mathbf{r}, \mathbf{r}') \mathbf{g}^n(\mathbf{r}', \boldsymbol{\alpha}) dr' \tag{55}
$$

At this point, it is convenient to make the following definitions:

$$\Phi(\mathbf{r}, \alpha \rho) = \mathcal{M} \, \prescript{m}{}{\mathbf{r}}(\mathbf{r}, \mathbf{r}\_0; \alpha \rho) - \Delta(\eta, \alpha \rho) \mathbf{g}^m(\mathbf{r}, \alpha \rho) \tag{56}$$

Fourier Transforms for Generalized Fredholm Equations DOI: http://dx.doi.org/10.5772/intechopen.85993

with also

Now, if we are near a resonance, Eqs. (48) and (49) are transformed in homogeneous equations with solutions we will denote as weð Þ ω , and if we denote the interaction as <sup>U</sup> and the kernel <sup>K</sup>ð Þ<sup>∘</sup> ð Þ <sup>ω</sup> , then from Eqs. (48) or (49) without the

This relation establishes that the resonant solutions are mutually orthogonal and

As we saw in Section 8, the resonant solutions are orthogonal and in Eq. (50) the Fredholm eigenvalues appear, but these last functions emerge when the inhomogeneous Fredholm equations are transformed in a homogeneous equation near a

According to the theory of homogeneous Fredholm equations [1, 2, 3, 5, 9, 15, 16],

one of the conditions for the existence of solutions is that first Fredholm's minor

From Eqs. (51) and (52) and after a little algebra, we arrive to the following

At this point, it is convenient to make the following definitions:

<sup>u</sup> � <sup>η</sup>�<sup>1</sup> l

<sup>¼</sup> <sup>0</sup> (52)

ð53Þ

ð54Þ

ð55Þ

ð56Þ

<sup>l</sup> ð Þ <sup>ω</sup> Uwuð Þ <sup>ω</sup> <sup>η</sup>�<sup>1</sup>

8. The homogeneous Fredholm equation and Fredholm's eigenvalue

source term, we have the following relation:

w†

Fourier Transforms - Century of Digitalization and Increasing Expectations

the functions η ωð Þ are known as the Fredholm eigenvalues.

resonance. The resulting homogeneous equation is

complies

equation:

34

$$\Phi^{(\circ)}(\mathbf{r};\boldsymbol{\alpha}) = $$

$$\Delta(\boldsymbol{\eta},\boldsymbol{\alpha})[\boldsymbol{\eta}(\boldsymbol{\alpha}) - \mathbf{g}^{\boldsymbol{m}(\circ)}(\mathbf{r},\boldsymbol{\alpha})]$$

$$+\Delta(\boldsymbol{\eta},\boldsymbol{\alpha})[\boldsymbol{\eta}(\boldsymbol{\alpha}) - \boldsymbol{\upsilon}(\boldsymbol{\alpha})]\Bigg[\overset{\circ}{\mathrm{K}}\_{\boldsymbol{\eta}}{\mathrm{K}}^{\mathrm{m}(\circ)}(\boldsymbol{\alpha},\mathbf{r},\mathbf{r}')\mathbf{g}^{\boldsymbol{\ast}}(\mathbf{r}',\boldsymbol{\alpha})dr'\tag{57}$$

We can reduce the last equations to a compact one:

$$\Phi(\mathbf{r}, \alpha) = $$

$$\Phi^{(\circ)}(\mathbf{r}, \alpha) + \eta(\alpha) \stackrel{\ast}{\int} \mathbf{K}\_{\eta}^{\eta(\circ)}(\alpha; \mathbf{r}, \mathbf{s}) \Phi(\mathbf{s}; \alpha) d\mathbf{s} \tag{58}$$

It is clear that our procedure leads to an inhomogeneous Fredholm equation in which it is possible to observe that the transit from a non-resonant regime to a resonant regime is described by the generalized source term <sup>Φ</sup>ð Þ<sup>∘</sup> ð Þ <sup>r</sup>;<sup>ω</sup> .

## 9. The role of resonances on broadcasting applications

In precedent sections we have seen how we can go from inhomogeneous to homogeneous Fredholm equations, that is, from non-resonant or conventional

#### Figure 1.

We show the superposition of three plasma layers subjected to local high electromagnetic potential creating resonances and releasing evanescent waves: Layer M is composed of magnetic plasma, and the U layers are composed of unmagnetized plasma.

solutions to resonant ones. But we know that the resonant solutions are related with a left-hand behavior of the transmitting media, that is, with negative refraction index. On the other hand, Xiang-kun Kong et al. [7, 11] have studied the sign change of the refraction index on devices with superposed layers of magnetized an unmagnetized plasma. This experiment suggested us to propose the plasma sandwich model for transmitting media illustrated in Figure 1 that consists in itinerant and random appearing of superposed magnetized and unmagnetized plasma layers in high atmosphere that creates localized zones with negative refraction index. According to the precedent results, the change to negative refraction index must establish completely different conditions for the crossing of electromagnetic signals, and we have the appropriate tool to handle these very important phenomena. That is we can observe the transition from evanescent waves (non-traveling waves) to traveling waves like an increase in the polarization effect. In Figure 1 three plasma regions appear named U (unmagnetized), M (magnetized), and U (again unmagnetized) representing a region on the atmosphere. When some local electromagnetic potential values occur, it is possible to reach left-hand material conditions. References

Press; 1958

309-324

703-725

37

York: Academic; 1965

York: Wiley; 1973

[1] Hoshtadt H. Integral Equations. New

DOI: http://dx.doi.org/10.5772/intechopen.85993

Fourier Transforms for Generalized Fredholm Equations

Applications (ICEAA), 2012 International Conference; 2-7 September 2012; Cape Town, South Africa: IEEE; pp. 392-395. DOI: 10.1109/

[11] Kong X-k, Liu S-b, Zhang H-f, Bian B-r, Li H-m, et al. Evanescent wave decomposition in a novel resonator comprising unmagnetized and magnetized plasma layers. Physics of Plasmas. 2013;20:043515. DOI: 10.1063/

[12] Velázquez-Arcos JM, Pérez-Martínez F, Rivera-Salamanca CA, Granados-Samaniego J. On the application of a recently discovered electromagnetic resonances to

om, ISSN: 2250-2459

Physics. 1996;37:4235

6046297

communication systems. International Journal of Emerging Technology and Advanced Engineering. 2013;3(1): 466-471. Available from: www.ijetae.c

[13] Bollini CG, Civitarese O, De Paoli AL, Rocca MC. Gamow states as continuous linear functionals over analytical test functions. Journal of Mathematical

[14] Velázquez-Arcos JM, Vargas CA, Fernández-Chapou JL, Granados-Samaniego J. Resonances on discrete electromagnetic time reversal applications. In: Electromagnetics in Advanced Applications (ICEAA), 2011 International Conference. 12-16 September 2011; Torino, Italy: IEEE; pp. 167-170. DOI: 10.1109/ICEAA.2011.

[15] Mondragón A, Hernández E, Velázquez-Arcos JM. Resonances and Gamow states in non-local potentials. Annalen der Physik. 1991;48:503-616. DOI: 10.1002/andp.19915030802

[16] Velázquez-Arcos JM, Vargas CA, Fernández-Chapou JL, Salas-Brito AL. On computing the trace of the kernel of

ICEAA.2012.6328657

1.4802807

[2] Mathews J, Walker RL. Mathematical

[4] Gradshteyn IS, Ryzhik IM. Tables of Integrals, Series and Products. New

[6] Grbic A, Eleftheriades GV. Negative refraction, growing evanescent waves, and sub-diffraction imaging in loaded transmission-line metamaterials. IEEE Transactions on Microwave Theory and Techniques. 2003;51(12):2297-20305

[7] Xu H-X, Wang G-M, Lv Y-Y, Qi M-Q, Gao X, Ge S. Multifrequency monopole antennas by loading metamaterial transmission lines with dual-shunt branch circuit. Progress in Electromagnetics Research. 2013;137:

[8] Hernández-Bautista F, Vargas CA, Velázquez-Arcos JM. Negative

refractive index in split ring resonators. Revista Mexicana de Fisica. 2013;59(1):

[9] Velázquez-Arcos JM. Fredholm's equations for subwavelength focusing. Journal of Mathematical Physics. 2012; 53(10):103520. DOI: 10.1063/1.4759502

[10] Velázquez-Arcos JM, Granados-

on electromagnetic systems. In: Electromagnetics in Advanced

Communication theory and resonances

139-144. ISSN: 0035-00IX

Samaniego J, Vargas CA.

Methods of Physics. Menlo Park, California: W.A. Benjamin, Inc.; 1973

[3] Smithies F. Integral Equations. Cambridge: Cambridge University

[5] von Der Heydt N. Schrödinger equation with non-local potential. I. The resolvent. Annalen der Physik. 1973;29:

## 10. Conclusions

In this chapter we have expanded the scope of Fourier transforms by application to a relatively new class (really a vector generalization) of integral equations we named generalized Fredholm equations (GFE). We think that the very relevant subjects we discussed, not only because they are far-reaching implications but also for they are not presented nowadays by other authors, are the properties we have discovered about both the GFE and its own solutions. We have shown a strong relation between the resonant solutions of the generalized homogeneous Fredholm equations for the electromagnetic field and the resonances observed in scattering in nuclear physics. The physical interpretation of the new class of resonances allows us to discern completely new applications in different subjects like electromagnetic wave propagation or the understanding of meta-materials. We give the mathematical proofs for properties of the integral equations, the relation between homogeneous and inhomogeneous equations, and the mechanism for release of the evanescent waves converting them in traveling ones.

## Author details

Juan Manuel Velazquez Arcos\*, Ricardo Teodoro Paez Hernandez, Alejandro Perez Ricardez and Jaime Granados Samaniego Universidad Autonoma Metropolitana, Mexico City, Mexico

\*Address all correspondence to: jmva@correo.azc.uam.mx

© 2019 The Author(s). Licensee IntechOpen. 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.

Fourier Transforms for Generalized Fredholm Equations DOI: http://dx.doi.org/10.5772/intechopen.85993

## References

solutions to resonant ones. But we know that the resonant solutions are related with a left-hand behavior of the transmitting media, that is, with negative refraction index. On the other hand, Xiang-kun Kong et al. [7, 11] have studied the sign change

of the refraction index on devices with superposed layers of magnetized an unmagnetized plasma. This experiment suggested us to propose the plasma sandwich model for transmitting media illustrated in Figure 1 that consists in itinerant and random appearing of superposed magnetized and unmagnetized plasma layers in high atmosphere that creates localized zones with negative refraction index. According to the precedent results, the change to negative refraction index must establish completely different conditions for the crossing of electromagnetic signals, and we have the appropriate tool to handle these very important phenomena. That is we can observe the transition from evanescent waves (non-traveling waves) to traveling waves like an increase in the polarization effect. In Figure 1 three plasma

Fourier Transforms - Century of Digitalization and Increasing Expectations

regions appear named U (unmagnetized), M (magnetized), and U (again

evanescent waves converting them in traveling ones.

Juan Manuel Velazquez Arcos\*, Ricardo Teodoro Paez Hernandez,

© 2019 The Author(s). Licensee IntechOpen. 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,

Alejandro Perez Ricardez and Jaime Granados Samaniego Universidad Autonoma Metropolitana, Mexico City, Mexico

\*Address all correspondence to: jmva@correo.azc.uam.mx

provided the original work is properly cited.

10. Conclusions

Author details

36

unmagnetized) representing a region on the atmosphere. When some local electromagnetic potential values occur, it is possible to reach left-hand material conditions.

In this chapter we have expanded the scope of Fourier transforms by application to a relatively new class (really a vector generalization) of integral equations we named generalized Fredholm equations (GFE). We think that the very relevant subjects we discussed, not only because they are far-reaching implications but also for they are not presented nowadays by other authors, are the properties we have discovered about both the GFE and its own solutions. We have shown a strong relation between the resonant solutions of the generalized homogeneous Fredholm equations for the electromagnetic field and the resonances observed in scattering in nuclear physics. The physical interpretation of the new class of resonances allows us to discern completely new applications in different subjects like electromagnetic wave propagation or the understanding of meta-materials. We give the mathematical proofs for properties of the integral equations, the relation between homogeneous and inhomogeneous equations, and the mechanism for release of the

[1] Hoshtadt H. Integral Equations. New York: Wiley; 1973

[2] Mathews J, Walker RL. Mathematical Methods of Physics. Menlo Park, California: W.A. Benjamin, Inc.; 1973

[3] Smithies F. Integral Equations. Cambridge: Cambridge University Press; 1958

[4] Gradshteyn IS, Ryzhik IM. Tables of Integrals, Series and Products. New York: Academic; 1965

[5] von Der Heydt N. Schrödinger equation with non-local potential. I. The resolvent. Annalen der Physik. 1973;29: 309-324

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[16] Velázquez-Arcos JM, Vargas CA, Fernández-Chapou JL, Salas-Brito AL. On computing the trace of the kernel of

Chapter 3

Abstract

1. Introduction

39

data, and it can be easily used online.

addition to the conventional RR interval analysis [1–5].

The RR Interval Spectrum, the

Alexander Gersten, Ori Gersten, Adi Ronen and Yair Cassuto

We discuss the relationship between the RR interval spectral analysis and the spectral analysis of the corresponding ECG signal from which the RR intervals were evaluated. The ECG signal spectrum is bounded below the frequency f<sup>B</sup> by using an electronic filter and sampled at rate larger than 2fB, thus excluding aliasing from spectral analysis. A similar procedure cannot be applied to the RR interval spectral analysis, and in this case aliasing is possible. One of our main efforts in this chapter is devoted to the problem of how to detect aliasing in the heart rate spectral analysis. In order to get an insight, we performed an experiment with an adult man, in which the ECG signal was detected in a case where the breathing rate was larger than half the heart rate. A constant breathing rate for time intervals exceeding 5 minutes was monitored with good accuracy using a special breathing procedure. The results show distinctively a very sharp peak in the spectral analysis of the ECG signal, and corresponding (diffused) aliasing peaks in the RR interval spectral analysis. A new method of dealing with unevenly sampled data was developed, which has interesting anti-aliasing properties. There are indications that the VLF peaks of the RR spectrum are originated by aliasing. Some of the LF peaks may have the same property. The chapter is fully based on the preprint arXiv:physics/9911017, submitted on 11 Nov 1999, by authors A. Gersten, O. Gersten, A. Ronen, and Y. Cassuto.

ECG Signal, and Aliasing

Keywords: heart rate, ECG signal, spectral analysis, aliasing, HRV

The RR interval spectral analysis is usually based on heart rate data collected in two ways. In one method, the data are collected by analog to digital conversion of the ECG signal and computer evaluation of the RR intervals from the ECG signal. In the second method, devices are used whose output is the RR interval alone. The advantage of the first method is the control of accuracy and flexibility of the evaluations. The second method has the advantage of storing smaller amount of

In the first method, usually the number of collected data (sampled ECG signal) is of two to three orders of magnitude larger than the RR interval data. Thus if only RR interval is analyzed, a large amount of data is unused. In this paper we are trying to take advantage of the ECG sampled signal and to derive new information in

The ECG signal spectrum is bounded below the frequency f<sup>B</sup> by using an electronic filter and sampled at rate larger than 2fB, thus excluding aliasing from

the homogeneous Fredholm's equation. Journal of Mathematical Physics. 2008; 49:103508. DOI: 10.1063/1.3003062

[17] de la Madrid R. The rigged Hilbert space approach to the Gamow states. Journal of Mathematical Physics. 2012; 53(10):102113. DOI: 10.1063/1.4758925

[18] Velázquez-Arcos JM, Granados-Samaniego J, Fernandez-Chapou JL, Vargas CA. Vector generalization of the discrete time reversal formalism brings an electromagnetic application on overcoming the diffraction limit. In: Electromagnetics in Advanced Applications (ICEAA), 2010 International Conference; 20-24 September 2010; Sydney, Australia: IEEE; pp. 264-267. DOI: 10.1109/ ICEAA.2010.5653059

[19] Kato H, Inoue M. Reflection-mode operation of one-dimensional magnetophotonic crystals for use in film-based magneto-optical isolator devices. Journal of Applied Physics. 2002;91:7017-7019

[20] Kato H, Matsushita T, Takayama A, Egawa M, Nishimura K, Inoue M. Theoretical analysis of optical and magneto-optical properties of onedimensional magnetophotonic crystals. Journal of Applied Physics. 2003;93: 3906

## Chapter 3

the homogeneous Fredholm's equation. Journal of Mathematical Physics. 2008; 49:103508. DOI: 10.1063/1.3003062

Fourier Transforms - Century of Digitalization and Increasing Expectations

[17] de la Madrid R. The rigged Hilbert space approach to the Gamow states. Journal of Mathematical Physics. 2012; 53(10):102113. DOI: 10.1063/1.4758925

[18] Velázquez-Arcos JM, Granados-Samaniego J, Fernandez-Chapou JL, Vargas CA. Vector generalization of the discrete time reversal formalism brings an electromagnetic application on overcoming the diffraction limit. In: Electromagnetics in Advanced Applications (ICEAA), 2010 International Conference; 20-24 September 2010; Sydney, Australia: IEEE; pp. 264-267. DOI: 10.1109/

[19] Kato H, Inoue M. Reflection-mode

[20] Kato H, Matsushita T, Takayama A, Egawa M, Nishimura K, Inoue M. Theoretical analysis of optical and magneto-optical properties of onedimensional magnetophotonic crystals. Journal of Applied Physics. 2003;93:

operation of one-dimensional magnetophotonic crystals for use in film-based magneto-optical isolator devices. Journal of Applied Physics.

ICEAA.2010.5653059

2002;91:7017-7019

3906

38
