**4. Conditions imposed over the homogeneous Fredholm equations**

In accordance with the theory of homogeneous Fredholm integral equations [1, 2, 13], the first Fredholm minor is a two point function, like a Green function, which must comply with an integral equation:

$$\begin{aligned} \odot \ell^{\mathfrak{m}}(\mathbf{r}, \mathbf{r}\_0; \boldsymbol{\alpha}) &= \eta(\boldsymbol{\alpha}) \Delta(\boldsymbol{\eta}, \boldsymbol{\alpha}) \\\\ &+ \eta(\boldsymbol{\alpha}) \int\_0^\infty \mathbf{K}\_n^{m(\circ)}(\boldsymbol{\alpha}; \mathbf{r}, \mathbf{s}) \odot \ell^n(\mathbf{s}, \mathbf{r}\_0; \boldsymbol{\alpha}) d\boldsymbol{s} \end{aligned} \tag{7}$$

Two other conditions must be satisfied: The first is that Fredholm determinant is zero

$$
\Delta(\eta, \alpha) = 0 \tag{8}
$$

The second that the Fredholm eigenvalue equals to one:

$$\eta(a) = \mathbf{1} \tag{9}$$

But thanks to our second order approximation Eq. (2) we can show that other interesting conditions are satisfied, for example if we define some particular functions (and operators):

$$\Psi(\mathbf{r};\alpha) \equiv \odot \ell^{\mathfrak{m}}(\mathbf{r},\mathbf{r}\_{0};\alpha) - \Delta(\eta,\alpha)u^{\mathfrak{m}}(\mathbf{r};\alpha) \tag{10}$$

And also

$$\begin{split} \Psi^{(\circ)}(\mathbf{r};\boldsymbol{\omega}) & \equiv \Delta(\boldsymbol{\eta},\boldsymbol{\omega}) \Big[ \boldsymbol{\eta}(\boldsymbol{\omega}) - \boldsymbol{\mu}^{m(\circ)}(\mathbf{r};\boldsymbol{\omega}) \Big] \\ &+ \Delta(\boldsymbol{\eta},\boldsymbol{\alpha}) [\boldsymbol{\eta}(\boldsymbol{\alpha}) - \boldsymbol{\nu}(\boldsymbol{\alpha})] \int \mathbf{K}\_{\boldsymbol{\eta}}^{m(\circ)}(\boldsymbol{\omega};\mathbf{r},\mathbf{r}') \boldsymbol{\mu}^{n}(\mathbf{r}';\boldsymbol{\omega}) d\boldsymbol{r}' \end{split} \tag{11}$$

The same operator (15) acting over a homogeneous equation looks like

þ Δð Þ*η* ½ � *η* � 0

*<sup>e</sup>* ð Þ¼ **r**; *ω*

*<sup>e</sup>* ð Þ¼ **<sup>r</sup>**;*<sup>ω</sup> <sup>λ</sup>ym*

<sup>Z</sup> **<sup>r</sup>**; *<sup>ω</sup>*; *<sup>u</sup><sup>m</sup>*ð Þ<sup>∘</sup> ð Þ **<sup>r</sup>**;*<sup>ω</sup>*

� � (18)

*<sup>i</sup> <sup>δ</sup>*ð Þ **<sup>r</sup>** � **<sup>r</sup>***<sup>i</sup>* **<sup>K</sup>***<sup>m</sup>*ð Þ<sup>∘</sup> *<sup>n</sup>* ð Þ *<sup>ω</sup>*; **<sup>r</sup>**, **<sup>r</sup>***<sup>i</sup>* (19)

!

*<sup>i</sup> <sup>δ</sup>*ð Þ **<sup>r</sup>** � **<sup>r</sup>***<sup>i</sup>* **<sup>K</sup>***<sup>m</sup>*ð Þ<sup>∘</sup> *<sup>n</sup>* <sup>ð</sup>*ω*; **<sup>r</sup>**, **<sup>r</sup>***i*<sup>Þ</sup>

� (20)

*<sup>e</sup>* ð Þ **r**; *ω* (21)

*<sup>e</sup>* **r**<sup>0</sup> ð Þ ;*ω dr*<sup>0</sup>

(16)

*<sup>e</sup>* ð Þ **r**;*ω* (17)

**<sup>K</sup>***<sup>m</sup>*ð Þ<sup>∘</sup> *<sup>n</sup> <sup>ω</sup>*; **<sup>r</sup>**, **<sup>r</sup>**<sup>0</sup> ð Þ*y<sup>n</sup>*

As we can see the effect of the Zap operator is to kill or eliminate the inhomogeneous term when applied to a resonant state. But this seems very artificial because we are giving indeed two parts for the complete rule. However we can build

On this section, we define the so named Zap projection operators (ZPO) which enable us to project a complex broadcasting system over a reduced resonant simplest one. The Zap operators acts over Fourier transforms [14, 15] related to integral

In order to get a display of the properties of this operator we propose a specific

Then, by applying the projection operator to Eq. (19) we have (remember that

<sup>½</sup>Δð Þ*<sup>η</sup> <sup>η</sup>* �<sup>X</sup>

**<sup>K</sup>***<sup>m</sup>*ð Þ<sup>∘</sup> *<sup>n</sup> <sup>ω</sup>*; **<sup>r</sup>**, **<sup>r</sup>**<sup>0</sup> ð Þ*un* **<sup>r</sup>**<sup>0</sup> ð Þ ; *<sup>ω</sup> dr*<sup>0</sup>

h i*um*ð Þ¼ **<sup>r</sup>**; *<sup>ω</sup> <sup>y</sup><sup>m</sup>*

*p*

*i*¼1 *αn*

Suppose that we have *p* punctual sources that can be represented in the

*p*

*i*¼1 *αn*

*<sup>e</sup>* ð Þ� **<sup>r</sup>**;*<sup>ω</sup>* <sup>Z</sup>½ � **<sup>r</sup>**; *<sup>ω</sup>*; 0 *<sup>y</sup><sup>m</sup>*

þ *η* ∞ð

*<sup>e</sup>* ð Þ� **<sup>r</sup>**; *<sup>ω</sup>* <sup>Z</sup>½ � **<sup>r</sup>**;*ω*; 0 *ym*

Now, we define Zap projection operators in the next section.

Then, based on (15) and (17), we define de following operator:

� � <sup>¼</sup> lim*<sup>η</sup>*!<sup>1</sup>

**6. The zap projection operators and their properties**

*<sup>P</sup>* **<sup>r</sup>**;*ω*; *<sup>u</sup><sup>m</sup>*ð Þ<sup>∘</sup> ð Þ **<sup>r</sup>**; *<sup>ω</sup>*

set of discrete antennas in the next example:

h i*um*ð Þ¼ **<sup>r</sup>**;*<sup>ω</sup>* lim*<sup>η</sup>*!<sup>1</sup>

when *η* ¼ 1 thus Δ ¼ 0):

*<sup>P</sup>* **<sup>r</sup>**;*ω*; *<sup>u</sup><sup>m</sup>*ð Þ<sup>∘</sup> ð Þ **<sup>r</sup>**; *<sup>ω</sup>*

**5**

inhomogeneous term of the Fredholm equation like:

*<sup>u</sup><sup>m</sup>*ð Þ<sup>∘</sup> ð Þ¼ **<sup>r</sup>**;*<sup>ω</sup>* <sup>X</sup>

þ*η* ∞ð

0

*<sup>P</sup>* **<sup>r</sup>**;*ω*; *<sup>u</sup><sup>m</sup>*ð Þ<sup>∘</sup> ð Þ **<sup>r</sup>**; *<sup>ω</sup>*

Now, because *η* ¼ 1 implies Δ ¼ 0, and because also *ν* ¼ *λ* ¼ 1

0

Z*y<sup>m</sup>*

*On the Zap Integral Operators over Fourier Transforms DOI: http://dx.doi.org/10.5772/intechopen.94573*

Z*y<sup>m</sup>*

projection operators that can make the work we need.

That is

operators.

We can see that the first Fredholm minor must satisfy through Ψ the inhomogeneous equation

$$\Psi(\mathbf{r};\boldsymbol{\alpha}) \equiv \Psi^{(\circ)}(\mathbf{r};\boldsymbol{\alpha}) + \eta(\boldsymbol{\alpha}) \int\_{0}^{\boldsymbol{\alpha}} \mathbf{K}\_{n}^{m(\circ)}(\boldsymbol{\alpha};\mathbf{r},\mathbf{t}) \Psi(\mathbf{t};\boldsymbol{\alpha}) d\mathbf{t} \tag{12}$$

In order to write Eq. (2) in terms of the solutions of Eq. (1), we can define the operator:

$$\Theta^{\mathfrak{m}}(\mathbf{r};\boldsymbol{\omega}) \equiv \nu^2(\boldsymbol{\alpha})\boldsymbol{\varepsilon}(\boldsymbol{\alpha}) \left[ \mathbf{K}\_{\boldsymbol{\alpha}}^{\mathfrak{m}(\boldsymbol{\ast})}(\boldsymbol{\alpha};\mathbf{r},\mathbf{r}') \right] \bigg|\_{\mbox{\tiny{\tiny}}\nolimits \mbox{\tiny{\tiny}}\nolimits \mbox{\tiny{\scriptsize{\textsc{\tiny}}}}{\mbox{\scriptsize{\textsc{\scriptsize{\textsc{\tiny}}}}}}(\boldsymbol{\alpha};\mathbf{r}',\mathbf{r}') \mathrm{P}^{\mathfrak{m}}(\mathbf{r}';\boldsymbol{\alpha}) dr' dr' \tag{13}$$

In Eq. (13) the function P*<sup>m</sup>*ð Þ **<sup>r</sup>**; *<sup>ω</sup>* is an arbitrary negative exponential regulator.

Near a resonance the two small parameters *ν ω*ð Þ and *ε ω*ð Þ makes <sup>Θ</sup>*<sup>m</sup>*ð Þ **<sup>r</sup>**; *<sup>ω</sup>* lesser than a second order term, so can be neglected. Far of a resonance this later function sketches the behavior of the simultaneous existence of the resonant and nonresonant solutions because in terms of <sup>Θ</sup>*<sup>m</sup>*ð Þ **<sup>r</sup>**; *<sup>ω</sup>* the conventional waves satisfy the inhomogeneous equation:

$$\begin{split} u^{m}(\mathbf{r};\boldsymbol{\alpha}) &= u^{m(\circ)}(\mathbf{r};\boldsymbol{\alpha}) + \Theta^{m}(\mathbf{r};\boldsymbol{\alpha}) \\ &+ \nu(\boldsymbol{\alpha}) \int\_{0}^{\boldsymbol{\alpha}} \mathbf{K}\_{n}^{m(\circ)}(\boldsymbol{\alpha};\mathbf{r},\mathbf{r}') u^{n}(\mathbf{r}';\boldsymbol{\alpha}) d\boldsymbol{r}' \end{split} \tag{14}$$

### **5. Defining a new class of integral operators**

As we said in Section **1**, hidden in the structure of Eq. (2) there are some integral operators which allow the simultaneous existence of solutions with extremely different boundary conditions. So, let us define the Zap operators by the rules:

$$\begin{aligned} \mathbf{Z}u^{m}(\mathbf{r};\boldsymbol{\alpha}) & \equiv \mathbf{Z} \Big[ \mathbf{r};\boldsymbol{\alpha};u^{m(\circ)}(\mathbf{r};\boldsymbol{\alpha}) \Big] u^{m}(\mathbf{r};\boldsymbol{\alpha}) = \\ & + \Delta(\boldsymbol{\eta}) \Big[ \boldsymbol{\eta} - u^{m(\boldsymbol{\alpha})}(\mathbf{r};\boldsymbol{\alpha}) \Big] \\ & + \eta \int\_{0}^{\boldsymbol{\alpha}} \mathbf{K}\_{n}^{m(\circ)}(\boldsymbol{\alpha};\mathbf{r},\mathbf{r}') u^{n}(\mathbf{r}';\boldsymbol{\alpha}) d\boldsymbol{r}' \end{aligned} \tag{15}$$

That is, the Zap operator is associated to the integral Fredholm equation satisfied by the affected solution (*um*ð Þ **<sup>r</sup>**;*<sup>ω</sup>* or *<sup>y</sup><sup>m</sup> <sup>e</sup>* ð Þ **r**; *ω* ), from which takes the source term and the free kernel.

The same operator (15) acting over a homogeneous equation looks like

$$\begin{aligned} \mathbf{Z} \mathbf{y}\_{\epsilon}^{m}(\mathbf{r}; \boldsymbol{\alpha}) & \equiv \mathbf{Z}[\mathbf{r}; \boldsymbol{\alpha}; \mathbf{0}] \mathbf{y}\_{\epsilon}^{m}(\mathbf{r}; \boldsymbol{\alpha}) = \\ &+ \Delta(\boldsymbol{\eta})[\boldsymbol{\eta} - \mathbf{0}] \\ &+ \eta \int\_{0}^{\boldsymbol{\alpha}} \mathbf{K}\_{n}^{m(\boldsymbol{\epsilon})}(\boldsymbol{\alpha}; \mathbf{r}, \mathbf{r}') \mathbf{y}\_{\epsilon}^{n}(\mathbf{r}'; \boldsymbol{\alpha}) d\mathbf{r}' \end{aligned} \tag{16}$$

That is

And also

geneous equation

<sup>Θ</sup>*<sup>m</sup>*ð Þ� **<sup>r</sup>**;*<sup>ω</sup> <sup>ν</sup>*<sup>2</sup>

inhomogeneous equation:

ð Þ *ω ε ω*ð Þ

ð ∞

**<sup>K</sup>***<sup>m</sup>*ð Þ<sup>∘</sup> *<sup>n</sup> <sup>ω</sup>*; **<sup>r</sup>**, **<sup>r</sup>**<sup>0</sup> ð Þ

*um*ð Þ¼ **<sup>r</sup>**;*<sup>ω</sup> <sup>u</sup><sup>m</sup>*ð Þ<sup>∘</sup> ð Þþ **<sup>r</sup>**;*<sup>ω</sup>* <sup>Θ</sup>*<sup>m</sup>*ð Þ **<sup>r</sup>**;*<sup>ω</sup>*

∞ð

0

operators which allow the simultaneous existence of solutions with extremely different boundary conditions. So, let us define the Zap operators by the rules:

<sup>Z</sup>*u<sup>m</sup>*ð Þ� **<sup>r</sup>**;*<sup>ω</sup>* <sup>Z</sup> **<sup>r</sup>**;*ω*; *<sup>u</sup><sup>m</sup>*ð Þ<sup>∘</sup> ð Þ **<sup>r</sup>**; *<sup>ω</sup>*

þ *η* ∞ð

0

As we said in Section **1**, hidden in the structure of Eq. (2) there are some integral

h i

<sup>þ</sup> <sup>Δ</sup>ð Þ*<sup>η</sup> <sup>η</sup>* � *um*ð Þ <sup>o</sup> ð Þ **<sup>r</sup>**; *<sup>ω</sup>* h i

That is, the Zap operator is associated to the integral Fredholm equation satisfied

**<sup>K</sup>***<sup>m</sup>*ð Þ<sup>∘</sup> *<sup>n</sup> <sup>ω</sup>*; **<sup>r</sup>**, **<sup>r</sup>**<sup>0</sup> ð Þ*u<sup>n</sup>* **<sup>r</sup>**<sup>0</sup> ð Þ ;*<sup>ω</sup> dr*<sup>0</sup>

þ *ν ω*ð Þ

**5. Defining a new class of integral operators**

by the affected solution (*um*ð Þ **<sup>r</sup>**;*<sup>ω</sup>* or *<sup>y</sup><sup>m</sup>*

and the free kernel.

**4**

0

operator:

<sup>Ψ</sup>ð Þ<sup>∘</sup> ð Þ� **<sup>r</sup>**;*<sup>ω</sup>* <sup>Δ</sup>ð Þ *<sup>η</sup>*, *<sup>ω</sup> η ω*ð Þ� *um*ð Þ<sup>∘</sup> <sup>ð</sup>**r**;*ω*<sup>Þ</sup>

þ Δð Þ *η*,*ω* ½ � *η ω*ð Þ� *ν ω*ð Þ

<sup>Ψ</sup>ð Þ� **<sup>r</sup>**;*<sup>ω</sup>* <sup>Ψ</sup>ð Þ<sup>∘</sup> ð Þþ **<sup>r</sup>**;*<sup>ω</sup> η ω*ð Þ

h i

*Real Perspective of Fourier Transforms and Current Developments in Superconductivity*

∞ð

**<sup>K</sup>***<sup>m</sup>*ð Þ<sup>∘</sup> *<sup>n</sup> <sup>ω</sup>*; **<sup>r</sup>**, **<sup>r</sup>**<sup>0</sup> ð Þ*un* **<sup>r</sup>**<sup>0</sup> ð Þ ; *<sup>ω</sup> dr*<sup>0</sup> (11)

**<sup>K</sup>***<sup>m</sup>*ð Þ<sup>∘</sup> *<sup>n</sup>* ð Þ *<sup>ω</sup>*; **<sup>r</sup>**,**<sup>t</sup>** <sup>Ψ</sup>ð Þ **<sup>t</sup>**; *<sup>ω</sup> dt* (12)

*dr*<sup>0</sup> (13)

(15)

, **<sup>r</sup>**<sup>00</sup> ð ÞP*<sup>m</sup>* **<sup>r</sup>**<sup>00</sup> ð Þ ; *<sup>ω</sup> dr*<sup>00</sup>

**<sup>K</sup>***<sup>m</sup>*ð Þ<sup>∘</sup> *<sup>n</sup> <sup>ω</sup>*; **<sup>r</sup>**, **<sup>r</sup>**<sup>0</sup> ð Þ*un* **<sup>r</sup>**<sup>0</sup> ð Þ ; *<sup>ω</sup> dr*<sup>0</sup> (14)

*um*ð Þ¼ **<sup>r</sup>**; *<sup>ω</sup>*

*<sup>e</sup>* ð Þ **r**; *ω* ), from which takes the source term

0

We can see that the first Fredholm minor must satisfy through Ψ the inhomo-

∞ð

0

In order to write Eq. (2) in terms of the solutions of Eq. (1), we can define the

∞ð

**K***<sup>n</sup>*ð Þ<sup>∘</sup> *<sup>l</sup> ω*; **r**<sup>0</sup>

0

In Eq. (13) the function P*<sup>m</sup>*ð Þ **<sup>r</sup>**; *<sup>ω</sup>* is an arbitrary negative exponential regulator. Near a resonance the two small parameters *ν ω*ð Þ and *ε ω*ð Þ makes <sup>Θ</sup>*<sup>m</sup>*ð Þ **<sup>r</sup>**; *<sup>ω</sup>* lesser than a second order term, so can be neglected. Far of a resonance this later function sketches the behavior of the simultaneous existence of the resonant and nonresonant solutions because in terms of <sup>Θ</sup>*<sup>m</sup>*ð Þ **<sup>r</sup>**; *<sup>ω</sup>* the conventional waves satisfy the

$$\mathbf{Z}\mathbf{y}\_{\varepsilon}^{m}(\mathbf{r};\alpha) \equiv \mathbf{Z}[\mathbf{r};\alpha;\mathbf{0}]\mathbf{y}\_{\varepsilon}^{m}(\mathbf{r};\alpha) = \lambda \mathbf{y}\_{\varepsilon}^{m}(\mathbf{r};\alpha) \tag{17}$$

As we can see the effect of the Zap operator is to kill or eliminate the inhomogeneous term when applied to a resonant state. But this seems very artificial because we are giving indeed two parts for the complete rule. However we can build projection operators that can make the work we need.

Now, we define Zap projection operators in the next section.

### **6. The zap projection operators and their properties**

On this section, we define the so named Zap projection operators (ZPO) which enable us to project a complex broadcasting system over a reduced resonant simplest one. The Zap operators acts over Fourier transforms [14, 15] related to integral operators.

Then, based on (15) and (17), we define de following operator:

$$\mathbb{Z}\_P\left(\mathbf{r}; \boldsymbol{\alpha}; \boldsymbol{u}^{m(\circ)}(\mathbf{r}; \boldsymbol{\alpha})\right) = \lim\_{\eta \to 1} \mathbb{Z}\left(\mathbf{r}; \boldsymbol{\alpha}; \boldsymbol{u}^{m(\circ)}(\mathbf{r}; \boldsymbol{\alpha})\right) \tag{18}$$

In order to get a display of the properties of this operator we propose a specific set of discrete antennas in the next example:

Suppose that we have *p* punctual sources that can be represented in the inhomogeneous term of the Fredholm equation like:

$$u^{m(\circ)}(\mathbf{r};w) = \sum\_{i=1}^{p} a\_i^n \delta(\mathbf{r} - \mathbf{r}\_i) \mathbf{K}\_n^{m(\circ)}(w; \mathbf{r}, \mathbf{r}\_i) \tag{19}$$

Then, by applying the projection operator to Eq. (19) we have (remember that when *η* ¼ 1 thus Δ ¼ 0):

$$\mathbb{Z}\_P\left[\mathbf{r};\,\boldsymbol{\alpha};\,\boldsymbol{u}^{m(\circ)}(\mathbf{r};\,\boldsymbol{\alpha})\right]\boldsymbol{u}^m(\mathbf{r};\,\boldsymbol{\omega})=\lim\_{\eta\to 1}\left[\Delta(\eta)\left(\eta-\sum\_{i=1}^p\alpha\_i^n\delta(\mathbf{r}-\mathbf{r}\_i)\mathbf{K}\_n^{m(\circ)}(\boldsymbol{\alpha};\,\mathbf{r},\mathbf{r}\_i)\right)\right]$$

$$+\eta\int\_0^\infty\mathbf{K}\_n^{m(\circ)}(\boldsymbol{\alpha};\,\mathbf{r},\mathbf{r}')\boldsymbol{u}^n(\mathbf{r}';\boldsymbol{\alpha})d\boldsymbol{r}'\right]\tag{20}$$

Now, because *η* ¼ 1 implies Δ ¼ 0, and because also *ν* ¼ *λ* ¼ 1

$$\mathbb{Z}\_P\Big[\mathbf{r};\,\boldsymbol{\alpha};\,\boldsymbol{u}^{m(\ast)}(\mathbf{r};\,\boldsymbol{\alpha})\Big]\boldsymbol{u}^m(\mathbf{r};\,\boldsymbol{\alpha})=\mathcal{Y}\_\epsilon^m(\mathbf{r};\,\boldsymbol{\alpha})\tag{21}$$

In the last step we have used the fact that the solution of the remaining homogeneous equation is denoted by *y<sup>m</sup> <sup>e</sup>* ð Þ **r**;*ω* .

Eq. (21) says that if we take a blend of regular and resonant solutions we have:

$$\mathbb{Z}\_P\left[\mathbf{r};\,\alpha;\,\mu^{m(\ast)}(\mathbf{r};\,\alpha)\right](\boldsymbol{u}^m(\mathbf{r};\,\alpha)+\boldsymbol{\mathcal{y}}\_\epsilon^m(\mathbf{r};\,\alpha))=\mathbf{2}\mathbf{y}\_\epsilon^m(\mathbf{r};\,\alpha)\tag{22}$$

In Eq. (25) *A*ð Þ**r** is the interaction that in the general case may contain a non-

*u*2ð Þ **r**;*ω* � � � **<sup>K</sup>**ð Þ<sup>∘</sup> ð Þ *<sup>ω</sup> <sup>g</sup>*1ð Þ *<sup>ω</sup>*

cos *ω* � *ω<sup>p</sup>* � �*d*

*ω* � *ω<sup>p</sup>* � �*d*

*<sup>α</sup>iδ*ð Þ **<sup>r</sup>** � **<sup>r</sup>***<sup>i</sup>* **<sup>K</sup>**ð Þ<sup>∘</sup> ð Þ *<sup>ω</sup>* **<sup>e</sup>***<sup>i</sup>* (28)

**<sup>K</sup>**ð Þ<sup>∘</sup> *<sup>ω</sup>*; **<sup>r</sup>**, **<sup>r</sup>**<sup>0</sup> ð Þ*<sup>u</sup>* **<sup>r</sup>**<sup>0</sup> ð Þ ,*<sup>ω</sup> dr*<sup>0</sup> (31)

1

CCCCA

*u*ð Þ**r** (33)

sin *ω* � *ω<sup>p</sup>* � �*d*

*ω* � *ω<sup>p</sup>* � �*d*

1

*<sup>u</sup>*ð Þ<sup>∘</sup> ð Þ¼ **<sup>r</sup>**; *<sup>ω</sup> <sup>α</sup>*1*δ*ð Þ **<sup>r</sup>** � **<sup>r</sup>**<sup>1</sup> **<sup>K</sup>**ð Þ<sup>∘</sup> ð Þ *<sup>ω</sup>* **<sup>e</sup>**<sup>1</sup> <sup>þ</sup> *<sup>α</sup>*2*δ*ð Þ **<sup>r</sup>** � **<sup>r</sup>**<sup>2</sup> **<sup>K</sup>**ð Þ<sup>∘</sup> ð Þ *<sup>ω</sup>* **<sup>e</sup>**<sup>2</sup> (30)

*g*2ð Þ *ω*

1

CCCA

� � (29)

(27)

(32)

� �*u*ð Þ**<sup>r</sup>** (26)

local potential, but not in our example.

*On the Zap Integral Operators over Fourier Transforms DOI: http://dx.doi.org/10.5772/intechopen.94573*

In Eq. (26) *u*ð Þ**r** is a scalar function. And then, the kernel may be

**<sup>K</sup>**ð Þ<sup>∘</sup> ð Þ¼ *<sup>ω</sup>*

<sup>11</sup> **<sup>K</sup>**ð Þ<sup>∘</sup> 12

*i*

*<sup>u</sup>*ð Þ<sup>∘</sup> ð Þ¼ **<sup>r</sup>**;*<sup>ω</sup>* <sup>X</sup>

**<sup>e</sup>**<sup>1</sup> <sup>¼</sup> <sup>1</sup> 0

The conventional waves satisfy the scalar form of Eq. (1)

*<sup>u</sup>*ð Þ¼ **<sup>r</sup>**, *<sup>ω</sup> <sup>u</sup>*ð Þ<sup>∘</sup> ð Þ **<sup>r</sup>**, *<sup>ω</sup>*

þ *λ ω*ð Þ

∞ð

0

*<sup>u</sup>*ð Þ¼ **<sup>r</sup>**, *<sup>ω</sup> <sup>u</sup>*ð Þ<sup>∘</sup> ð Þ **<sup>r</sup>**, *<sup>ω</sup>*

Where the form of *u*ð Þ **r**,*ω* is unknown but possibly be sketched as

0

BBBB@

*u*ð Þ¼ **r**,*ω*

<sup>þ</sup> *λ ω*ð Þ**K**ð Þ<sup>∘</sup> ð Þ *<sup>ω</sup> <sup>u</sup>*ð Þ **<sup>r</sup>**, *<sup>ω</sup>*

sin *ω* � *ω<sup>p</sup>* � �*d*

*ω* � *ω<sup>p</sup>* � �*d* sin *<sup>ω</sup>* � *<sup>ω</sup><sup>p</sup>* <sup>þ</sup> *<sup>β</sup>* � �*<sup>d</sup> <sup>ω</sup>* � *<sup>ω</sup><sup>p</sup>* <sup>þ</sup> *<sup>β</sup>* � �*<sup>d</sup>*

Now we can apply the projection operator *<sup>P</sup>* **<sup>r</sup>**; *<sup>ω</sup>*; *<sup>u</sup>*ð Þ<sup>∘</sup> ð Þ **<sup>r</sup>**;*<sup>ω</sup>* � � to Eq. (32) and

0

BBB@

! *<sup>u</sup>*1ð Þ **<sup>r</sup>**; *<sup>ω</sup>*

sin *ω* � *ω<sup>p</sup>* � �*d*

*ω* � *ω<sup>p</sup>* � �*<sup>d</sup>* �*<sup>i</sup>*

cos *ω* � *ω<sup>p</sup>* � �*d*

*ω* � *ω<sup>p</sup>* � �*d*

2

*i*¼1

� � and**e**<sup>2</sup> <sup>¼</sup> <sup>0</sup>

**K**ð Þ<sup>∘</sup> <sup>21</sup> **<sup>K</sup>**ð Þ<sup>∘</sup> 22

**<sup>K</sup>**ð Þ<sup>∘</sup> ð Þ *<sup>ω</sup> <sup>u</sup>*ð Þ� **<sup>r</sup>**; *<sup>ω</sup>* **<sup>K</sup>**ð Þ<sup>∘</sup>

So Eq. (19) takes the form:

Or in accordance with Eq. (25)

Where

That is

obtain

**7**

So taken into account from Eqs. (18) until (22), we see that we have projected the original problem into a resonant one.

In analogy with *<sup>P</sup>* we can define a projector over their complement: Let us define the complementary Zap projection operator as

$$\begin{split} \mathbb{Z}\_{Q}\left(\mathbf{r};\boldsymbol{\omega};\boldsymbol{u}^{m(\circ)}(\mathbf{r};\boldsymbol{\omega})\right)\boldsymbol{u}^{m}(\mathbf{r};\boldsymbol{\omega}) & \equiv \lim\_{\eta \to 1} \boldsymbol{Z}^{C}\Big(\mathbf{r};\boldsymbol{\omega};\boldsymbol{u}^{m(\circ)}(\mathbf{r};\boldsymbol{\omega})\Big) \\ & \equiv \lim\_{\eta \to 1} \left\{ (\Delta(\eta)\eta + \nu) \int\_{0}^{\infty} \mathbf{K}\_{\boldsymbol{u}}^{m(\circ)}(\boldsymbol{\omega};\mathbf{r},\mathbf{r}')\boldsymbol{u}^{n}(\mathbf{r}';\boldsymbol{\omega}) d\boldsymbol{r}' \right. \\ & \left. + \eta \boldsymbol{u}^{m(\circ)}(\mathbf{r};\boldsymbol{\omega})\right\} = \boldsymbol{u}^{m}(\mathbf{r};\boldsymbol{\omega}) \end{split} \tag{23}$$

Even we apply *<sup>Q</sup>* to a resonant state:

$$\begin{split} \mathbb{Z}\_{\mathcal{Q}}\left(\mathbf{r};\boldsymbol{\omega};\boldsymbol{u}^{m(\circ)}(\mathbf{r};\boldsymbol{\omega})\right)\boldsymbol{\uprho}\_{\epsilon}^{m}(\mathbf{r};\boldsymbol{\omega}) &\equiv \lim\_{\eta\to 1} \boldsymbol{Z}^{C}\Big(\mathbf{r};\boldsymbol{\omega};\boldsymbol{u}^{m(\circ)}(\mathbf{r};\boldsymbol{\omega})\Big) \\ &= \lim\_{\eta\to 1} \left\{ (\boldsymbol{\Delta}(\eta)\boldsymbol{\eta}+\boldsymbol{\nu}) \int\_{0}^{\infty} \mathbf{K}\_{n}^{m(\circ)}(\boldsymbol{\omega};\mathbf{r},\mathbf{r}')\boldsymbol{\uprho}\_{\epsilon}^{n}(\mathbf{r}';\boldsymbol{\omega})d\boldsymbol{r}' \right. \\ &\left. + \boldsymbol{\eta}\boldsymbol{u}^{m(\circ)}(\mathbf{r};\boldsymbol{\omega})\right\} = \boldsymbol{u}^{m}(\mathbf{r};\boldsymbol{\omega}) \end{split} \tag{24}$$

This is because the name of the solution of the remaining inhomogeneous equation is precisely *um*ð Þ **<sup>r</sup>**; *<sup>ω</sup>* .

### **7. An academic example for conventional traveling waves**

In order to convince us of the utility of the *<sup>P</sup>* and *<sup>Q</sup>* operators we remember that in all of our developments the kernel always is **K***<sup>m</sup>*ð Þ<sup>∘</sup> *<sup>n</sup>* that only contains the free Green function **G***<sup>t</sup>*ð Þ<sup>∘</sup> *<sup>n</sup>* ð Þ *ω*; **r**, **s .** But then, there is no difference between the kernels of the integral equations when are referred to conventional traveling waves or to evanescent or resonant waves. This last statement allows describing in an algebraic mode the application of the Zap projection operators. In this manner we can fix our kernel in accordance with a previous example that we have presented in some place as the matrix (27).

For the case of only two source points and omitting the three components of the field lifting only one, this matrix can be for example:

But first remember that

$$\int\_{V} A(\mathbf{r}')G\_{\boldsymbol{\alpha}}^{(\circ)}(\mathbf{r};\mathbf{r}')u(\mathbf{r}';\boldsymbol{\alpha})dV' \equiv \mathbf{K}^{(\circ)}(\boldsymbol{\alpha})u(\mathbf{r};\boldsymbol{\alpha})\tag{25}$$

In Eq. (25) *A*ð Þ**r** is the interaction that in the general case may contain a nonlocal potential, but not in our example.

$$\mathbf{K}^{(\circ)}(w)u(\mathbf{r};w) \equiv \begin{pmatrix} \mathbf{K}\_{11}^{(\circ)} & \mathbf{K}\_{12}^{(\circ)} \\ \mathbf{K}\_{21}^{(\circ)} & \mathbf{K}\_{22}^{(\circ)} \end{pmatrix} \begin{pmatrix} u\_1(\mathbf{r};w) \\ u\_2(\mathbf{r};w) \end{pmatrix} \equiv \mathbf{K}^{(\circ)}(w) \begin{pmatrix} g\_1(w) \\ g\_2(w) \end{pmatrix} u(\mathbf{r}) \tag{26}$$

In Eq. (26) *u*ð Þ**r** is a scalar function. And then, the kernel may be

$$\mathbf{K}^{(\circ)}(\boldsymbol{\omega}) = \begin{pmatrix} \frac{\sin\left(\boldsymbol{\omega} - \boldsymbol{\alpha}\_{p}\right)d}{\left(\boldsymbol{\omega} - \boldsymbol{\alpha}\_{p}\right)d} & -i\frac{\cos\left(\boldsymbol{\omega} - \boldsymbol{\alpha}\_{p}\right)d}{\left(\boldsymbol{\omega} - \boldsymbol{\alpha}\_{p}\right)d} \\ i\frac{\cos\left(\boldsymbol{\omega} - \boldsymbol{\alpha}\_{p}\right)d}{\left(\boldsymbol{\omega} - \boldsymbol{\alpha}\_{p}\right)d} & \frac{\sin\left(\boldsymbol{\omega} - \boldsymbol{\alpha}\_{p}\right)d}{\left(\boldsymbol{\omega} - \boldsymbol{\alpha}\_{p}\right)d} \end{pmatrix} \tag{27}$$

So Eq. (19) takes the form:

$$u^{(\circ)}(\mathbf{r};\boldsymbol{\alpha}) = \sum\_{i=1}^{2} a\_i \delta(\mathbf{r} - \mathbf{r}\_i) \mathbf{K}^{(\circ)}(\boldsymbol{\alpha}) \mathbf{e}\_i \tag{28}$$

Where

In the last step we have used the fact that the solution of the remaining homo-

Eq. (21) says that if we take a blend of regular and resonant solutions we have:

<sup>ð</sup>*um*ð Þþ **<sup>r</sup>**; *<sup>ω</sup> <sup>y</sup><sup>m</sup>*

So taken into account from Eqs. (18) until (22), we see that we have projected

*<sup>e</sup>* ð ÞÞ ¼ **<sup>r</sup>**;*<sup>ω</sup>* <sup>2</sup>*y<sup>m</sup>*

<sup>Z</sup>*<sup>C</sup>* **<sup>r</sup>**; *<sup>ω</sup>*; *<sup>u</sup>m*ð Þ<sup>∘</sup> ð Þ **<sup>r</sup>**;*<sup>ω</sup>* � �

<sup>Z</sup>*<sup>C</sup>* **<sup>r</sup>**; *<sup>ω</sup>*; *<sup>u</sup><sup>m</sup>*ð Þ<sup>∘</sup> ð Þ **<sup>r</sup>**;*<sup>ω</sup>* � �

ð ∞

0

ð ∞

**<sup>K</sup>***<sup>m</sup>*ð Þ<sup>∘</sup> *<sup>n</sup> <sup>ω</sup>*; **<sup>r</sup>**, **<sup>r</sup>**<sup>0</sup> ð Þ*y<sup>n</sup>*

0

fð Þ Δð Þ*η η* þ *ν*

<sup>þ</sup> *<sup>η</sup>um*ð Þ<sup>∘</sup> ð Þg ¼ **<sup>r</sup>**;*<sup>ω</sup> um*ð Þ **<sup>r</sup>**; *<sup>ω</sup>*

fð Þ Δð Þ*η η* þ *ν*

<sup>þ</sup> *<sup>η</sup>um*ð Þ<sup>∘</sup> ð Þg ¼ **<sup>r</sup>**;*<sup>ω</sup> um*ð Þ **<sup>r</sup>**; *<sup>ω</sup>*

This is because the name of the solution of the remaining inhomogeneous equa-

In order to convince us of the utility of the *<sup>P</sup>* and *<sup>Q</sup>* operators we remember that in all of our developments the kernel always is **K***<sup>m</sup>*ð Þ<sup>∘</sup> *<sup>n</sup>* that only contains the free

For the case of only two source points and omitting the three components of the

the integral equations when are referred to conventional traveling waves or to evanescent or resonant waves. This last statement allows describing in an algebraic mode the application of the Zap projection operators. In this manner we can fix our kernel in accordance with a previous example that we have presented in some place

*<sup>n</sup>* ð Þ *ω*; **r**, **s .** But then, there is no difference between the kernels of

*<sup>ω</sup>* **<sup>r</sup>**; **<sup>r</sup>**<sup>0</sup> ð Þ*<sup>u</sup>* **<sup>r</sup>**<sup>0</sup> ð Þ ; *<sup>ω</sup> dV*<sup>0</sup> � **<sup>K</sup>**ð Þ<sup>∘</sup> ð Þ *<sup>ω</sup> <sup>u</sup>*ð Þ **<sup>r</sup>**; *<sup>ω</sup>* (25)

*<sup>e</sup>* ð Þ **r**; *ω* (22)

**<sup>K</sup>***<sup>m</sup>*ð Þ<sup>∘</sup> *<sup>n</sup> <sup>ω</sup>*; **<sup>r</sup>**, **<sup>r</sup>**<sup>0</sup> ð Þ*u<sup>n</sup>* **<sup>r</sup>**<sup>0</sup> ð Þ ;*<sup>ω</sup> dr*<sup>0</sup>

(23)

(24)

*<sup>e</sup>* **r**<sup>0</sup> ð Þ ; *ω dr*<sup>0</sup>

*<sup>e</sup>* ð Þ **r**;*ω* .

*Real Perspective of Fourier Transforms and Current Developments in Superconductivity*

In analogy with *<sup>P</sup>* we can define a projector over their complement:

� lim*<sup>η</sup>*!<sup>1</sup>

Let us define the complementary Zap projection operator as

*um*ð Þ� **<sup>r</sup>**; *<sup>ω</sup>* lim*<sup>η</sup>*!<sup>1</sup>

*<sup>e</sup>* ð Þ� **<sup>r</sup>**; *<sup>ω</sup>* lim*<sup>η</sup>*!<sup>1</sup>

**7. An academic example for conventional traveling waves**

field lifting only one, this matrix can be for example:

*<sup>A</sup>* **<sup>r</sup>**<sup>0</sup> ð Þ*G*ð Þ<sup>∘</sup>

<sup>¼</sup> lim*<sup>η</sup>*!<sup>1</sup>

geneous equation is denoted by *y<sup>m</sup>*

*<sup>P</sup>* **<sup>r</sup>**; *<sup>ω</sup>*; *<sup>u</sup>m*ð Þ<sup>∘</sup> ð Þ **<sup>r</sup>**; *<sup>ω</sup>* h i

the original problem into a resonant one.

Even we apply *<sup>Q</sup>* to a resonant state:

*ym*

*<sup>Q</sup>* **<sup>r</sup>**;*ω*; *<sup>u</sup>m*ð Þ<sup>∘</sup> ð Þ **<sup>r</sup>**; *<sup>ω</sup>* � �

*<sup>Q</sup>* **<sup>r</sup>**; *<sup>ω</sup>*; *<sup>u</sup><sup>m</sup>*ð Þ<sup>∘</sup> ð Þ **<sup>r</sup>**;*<sup>ω</sup>* � �

tion is precisely *um*ð Þ **<sup>r</sup>**; *<sup>ω</sup>* .

Green function **G***<sup>t</sup>*ð Þ<sup>∘</sup>

as the matrix (27).

**6**

But first remember that

ð

*V*

$$\mathbf{e}\_1 = \begin{pmatrix} \mathbf{1} \\ \mathbf{0} \end{pmatrix} \text{ and} \mathbf{e}\_2 = \begin{pmatrix} \mathbf{0} \\ \mathbf{1} \end{pmatrix} \tag{29}$$

That is

$$u^{(\ast)}(\mathbf{r};\boldsymbol{\alpha}) = a\_1 \delta(\mathbf{r} - \mathbf{r}\_1) \mathbf{K}^{(\ast)}(\boldsymbol{\alpha}) \mathbf{e}\_1 + a\_2 \delta(\mathbf{r} - \mathbf{r}\_2) \mathbf{K}^{(\ast)}(\boldsymbol{\alpha}) \mathbf{e}\_2 \tag{30}$$

The conventional waves satisfy the scalar form of Eq. (1)

$$\begin{aligned} u(\mathbf{r}, \omega) &= u^{(\circ)}(\mathbf{r}, \omega) \\ &+ \lambda(\omega) \int\_0^\infty \mathbf{K}^{(\circ)}(\boldsymbol{\omega}; \mathbf{r}, \mathbf{r}') u(\mathbf{r}', \omega) d\boldsymbol{r}' \end{aligned} \tag{31}$$

Or in accordance with Eq. (25)

$$\begin{aligned} u(\mathbf{r}, a) &= u^{(\circ)}(\mathbf{r}, a) \\ &+ \lambda(a) \mathbf{K}^{(\circ)}(a) u(\mathbf{r}, a) \end{aligned} \tag{32}$$

Where the form of *u*ð Þ **r**,*ω* is unknown but possibly be sketched as

$$u(\mathbf{r}, \omega) = \begin{pmatrix} \frac{\sin\left(\omega - \alpha\_p\right)d}{(\alpha - \alpha\_p)d} \\\\ \frac{\sin\left(\omega - \alpha\_p + \beta\right)d}{(\alpha - \alpha\_p + \beta)d} \end{pmatrix} u(\mathbf{r}) \tag{33}$$

Now we can apply the projection operator *<sup>P</sup>* **<sup>r</sup>**; *<sup>ω</sup>*; *<sup>u</sup>*ð Þ<sup>∘</sup> ð Þ **<sup>r</sup>**;*<sup>ω</sup>* � � to Eq. (32) and obtain

*Real Perspective of Fourier Transforms and Current Developments in Superconductivity*

$$\mathbb{Z}\_P\left[\mathbf{r};\boldsymbol{\omega};\boldsymbol{\upmu}^{(\circ)}(\mathbf{r};\boldsymbol{\upmu})\right]\boldsymbol{\upmu}(\mathbf{r};\boldsymbol{\upmu})=\lim\_{\eta\to 1}\left[\boldsymbol{\upLambda}(\eta)\left(\eta-\sum\_{i=1}^2a\_i\delta(\mathbf{r}-\mathbf{r}\_i)\mathbf{K}^{(\circ)}(\boldsymbol{\upmu})\mathbf{e}\_i\right)\right]$$

$$+\boldsymbol{\upmu}\mathbf{K}^{(\circ)}(\boldsymbol{\upmu})\boldsymbol{\upmu}(\mathbf{r};\boldsymbol{\upmu})\tag{34}$$

Then, by putting *η* ¼ 1 and Δ ¼ 0 finally

$$\mathbb{Z}\_P\left[\mathbf{r};\,\alpha;\,\mathfrak{u}^{(\ast)}(\mathbf{r};\,\alpha)\right]\mathfrak{u}(\mathbf{r};\,\alpha) = \mathfrak{y}\_\epsilon(\mathbf{r};\,\alpha) \tag{35}$$

So it is irrelevant the part of the problem concerning the two sources, it is only a problem about resonances. Our problem is now to find the resonant frequencies by taking **<sup>K</sup>**ð Þ<sup>∘</sup> ð Þ *<sup>ω</sup>* and impose the conditions *<sup>η</sup>* <sup>¼</sup> 1 and <sup>Δ</sup> <sup>¼</sup> <sup>0</sup>**.**

But, what is the real advantage of the *<sup>P</sup>* and *<sup>Q</sup>* operators?, the answer is that the Zap operator formalism may be viewed as a test for distinguish between an expression that cannot be transformed or yes, in whatever sense between the homogeneous and inhomogeneous equations under the rules established above; if not, we can ensure that some kind of irregular things are present. In case of the positive transformation, we have the confidence that both kinds of solutions can coexists, and then we can separate the solutions for convenience as if it was a problem of two steps: homogeneous and inhomogeneous.

Now the last condition over the Fredholm determinant is

$$
\Delta \begin{pmatrix}
\frac{\sin\left(\omega - \alpha\_p\right)d}{\left(\omega - \alpha\_p\right)d} - \eta & -i \frac{\cos\left(\omega - \alpha\_p\right)d}{\left(\omega - \alpha\_p\right)d} \\
\frac{\cos\left(\omega - \alpha\_p\right)d}{\left(\omega - \alpha\_p\right)d} & \frac{\sin\left(\omega - \alpha\_p\right)d}{\left(\omega - \alpha\_p\right)d} - \eta
\end{pmatrix} = 0
\tag{36}
$$

The parameters *d*, *η* **(**the Fredholm eigenvalue) and *ω<sup>p</sup>* (the plasma frequency) can take in principle, arbitrary values but for a specific media can be take numeric values. Now, we remember that we must also impose *η* ¼ 1**.**

Then, Eq. (36) has two resonances:

$$
\alpha\_1 = \frac{\pi}{4d} + \alpha\_p \tag{37}
$$

Ωð Þ <sup>þ</sup>

Ωð Þ �

C*<sup>n</sup>*,*<sup>e</sup>* 2*ω<sup>e</sup>* j j*p*4*πω<sup>e</sup>*

*<sup>i</sup>ωet* ∗ P*<sup>e</sup> Sb* ð Þ*t e*

ð Þ 2ð Þ *ω* � *ω<sup>e</sup> ω<sup>e</sup>* is a rectangular function.

*Sa*

*<sup>e</sup>*,*<sup>T</sup> Sa*ðÞ¼ *t e*

And

**Figure 1.**

*F* P*<sup>e</sup> Sa* ð Þ*t e <sup>i</sup>ωet* h i <sup>¼</sup> <sup>X</sup><sup>∞</sup>

*ym*

**9**

Where *p*4*πω<sup>e</sup>*

**9. Conclusions**

In Eqs. (39) and (40) P*<sup>e</sup>*

*The two rectangular functions p*4*πω<sup>e</sup>*

Denoting the Fourier transform like

*On the Zap Integral Operators over Fourier Transforms DOI: http://dx.doi.org/10.5772/intechopen.94573*

*n*¼�∞

And for the convolution we have

*F* P*<sup>e</sup> Sa* ð Þ*t e*

runners of the Zap projection operators.

Then, the Fourier transform of the GPO is

*<sup>e</sup>*,*<sup>T</sup> Sa*ð Þ¼ *t* � *T e*

ð Þ 2ð Þ *ω* � *ω<sup>e</sup> ω<sup>e</sup> and p*4*πω<sup>u</sup>*

*iωet* P*e Sa*

ð Þ 2ð Þ *ω* � *ω<sup>u</sup> ω<sup>u</sup> :*

ð Þ *T* � *t* are simple projection operators [11].

*<sup>i</sup>ωut* h i <sup>¼</sup> *FSa* ð Þ *<sup>ω</sup>* � *<sup>ω</sup><sup>e</sup> FSb* ð Þ *<sup>ω</sup>* � *<sup>ω</sup><sup>u</sup>* (43)

*F*½ �� *f t*ð Þ Fð Þ *ω* (41)

*<sup>i</sup>ωe*ð Þ *<sup>T</sup>*�*<sup>t</sup>* P*<sup>e</sup> Sa*

ð Þ 2ð Þ *ω* � *ω<sup>e</sup> ω<sup>e</sup> e*

Then we see that the set of Fourier transforms of the GPOs behaves like a set of orthogonal basis functions for the frequency domain, that is, the resonant functions

*<sup>e</sup>* ð Þ **r**;*ω* as we can verify in **Figure 1**. So the GPO can be considered as the fore-

We can conclude that the Zap projection operators (ZPO) can be used as an alternative approach to the generalized projection operators (GPO) that is like an alternative for clean the evanescent signals [10] from disturbances generated by the sources and at the same time to clean the source signals from resonant solutions. We

ð Þ*t* (39)

ð Þ *T* � *t* (40)

�*i*ð Þ *<sup>ω</sup>*�*ω<sup>e</sup> <sup>π</sup><sup>n</sup>* � *FSa* ð Þ *<sup>ω</sup>* � *<sup>ω</sup><sup>e</sup>* (42)

And

$$
\alpha\_1 = \frac{3\pi}{4d} + \alpha\_p \tag{38}
$$

### **8. Forerunners of the zap projection operators**

In Section **6** we defined a new class of integral operators we named Zap projection operators that literally cleans from a broadcasting problem the inhomogeneous part and leaves a projected homogeneous version. These operators act directly over an inhomogeneous Fredholm equation and are related to the Fredholm operators. But recently, we have defined another set of operators we called generalized projection operators (GPO) which projects a complete broadcasting signal (maybe described by a GIFE) not only into several independent mutually orthogonal signals but also can reverse the time direction as we wish. These GPO may be considered as the precursors of the Zap operators and we will see why. We remember their form:

*On the Zap Integral Operators over Fourier Transforms DOI: http://dx.doi.org/10.5772/intechopen.94573*

**Figure 1.** *The two rectangular functions p*4*πω<sup>e</sup>* ð Þ 2ð Þ *ω* � *ω<sup>e</sup> ω<sup>e</sup> and p*4*πω<sup>u</sup>* ð Þ 2ð Þ *ω* � *ω<sup>u</sup> ω<sup>u</sup> :*

$$\mathfrak{Q}\_{e,T}^{(+)}\mathbb{S}\_{a}(t-T) = e^{i\alpha\_{t}t}\mathbb{P}\_{\mathbb{S}\_{a}}^{\epsilon}(t) \tag{39}$$

And

*<sup>P</sup>* **<sup>r</sup>**;*ω*; *<sup>u</sup>*ð Þ<sup>∘</sup> ð Þ **<sup>r</sup>**;*<sup>ω</sup>* h i

Then, by putting *η* ¼ 1 and Δ ¼ 0 finally

*<sup>u</sup>*ð Þ¼ **<sup>r</sup>**;*<sup>ω</sup>* lim*<sup>η</sup>*!<sup>1</sup>

*Real Perspective of Fourier Transforms and Current Developments in Superconductivity*

*<sup>P</sup>* **<sup>r</sup>**; *<sup>ω</sup>*; *<sup>u</sup>*ð Þ<sup>∘</sup> ð Þ **<sup>r</sup>**; *<sup>ω</sup>* h i

taking **<sup>K</sup>**ð Þ<sup>∘</sup> ð Þ *<sup>ω</sup>* and impose the conditions *<sup>η</sup>* <sup>¼</sup> 1 and <sup>Δ</sup> <sup>¼</sup> <sup>0</sup>**.**

problem of two steps: homogeneous and inhomogeneous. Now the last condition over the Fredholm determinant is

> sin *ω* � *ω<sup>p</sup>* � �*d*

> > *ω* � *ω<sup>p</sup>*

cos *ω* � *ω<sup>p</sup>* � �*d*

*ω* � *ω<sup>p</sup>* � �*d*

values. Now, we remember that we must also impose *η* ¼ 1**.**

**8. Forerunners of the zap projection operators**

� �*<sup>d</sup>* � *<sup>η</sup>* �*<sup>i</sup>*

Δ

0

BBB@

Then, Eq. (36) has two resonances:

And

**8**

*i*

<sup>½</sup>Δð Þ*<sup>η</sup> <sup>η</sup>* �<sup>X</sup>

So it is irrelevant the part of the problem concerning the two sources, it is only a problem about resonances. Our problem is now to find the resonant frequencies by

But, what is the real advantage of the *<sup>P</sup>* and *<sup>Q</sup>* operators?, the answer is that the Zap operator formalism may be viewed as a test for distinguish between an expression that cannot be transformed or yes, in whatever sense between the homogeneous and inhomogeneous equations under the rules established above; if not, we can ensure that some kind of irregular things are present. In case of the positive transformation, we have the confidence that both kinds of solutions can coexists, and then we can separate the solutions for convenience as if it was a

2

*<sup>α</sup>iδ*ð Þ **<sup>r</sup>** � **<sup>r</sup>***<sup>i</sup>* **<sup>K</sup>**ð Þ<sup>∘</sup> ð Þ *<sup>ω</sup>* **<sup>e</sup>***<sup>i</sup>*

!

*u*ð Þ¼ **r**;*ω ye*ð Þ **r**;*ω* (35)

1

<sup>4</sup>*<sup>d</sup>* <sup>þ</sup> *<sup>ω</sup><sup>p</sup>* (37)

<sup>4</sup>*<sup>d</sup>* <sup>þ</sup> *<sup>ω</sup><sup>p</sup>* (38)

CCCA <sup>¼</sup> 0 (36)

<sup>þ</sup>*η***K**ð Þ<sup>∘</sup> ð Þ *<sup>ω</sup> <sup>u</sup>*ð Þ� **<sup>r</sup>**;*<sup>ω</sup>* (34)

*i*¼1

cos *ω* � *ω<sup>p</sup>* � �*d*

*ω* � *ω<sup>p</sup>* � �*d*

sin *ω* � *ω<sup>p</sup>* � �*d*

*ω* � *ω<sup>p</sup>* � �*<sup>d</sup>* � *<sup>η</sup>*

The parameters *d*, *η* **(**the Fredholm eigenvalue) and *ω<sup>p</sup>* (the plasma frequency) can take in principle, arbitrary values but for a specific media can be take numeric

*<sup>ω</sup>*<sup>1</sup> <sup>¼</sup> *<sup>π</sup>*

*<sup>ω</sup>*<sup>1</sup> <sup>¼</sup> <sup>3</sup>*<sup>π</sup>*

In Section **6** we defined a new class of integral operators we named Zap projection operators that literally cleans from a broadcasting problem the inhomogeneous part and leaves a projected homogeneous version. These operators act directly over an inhomogeneous Fredholm equation and are related to the Fredholm operators. But recently, we have defined another set of operators we called generalized projection operators (GPO) which projects a complete broadcasting signal (maybe described by a GIFE) not only into several independent mutually orthogonal signals but also can reverse the time direction as we wish. These GPO may be considered as the precursors of the Zap operators and we will see why. We remember their form:

$$\mathbf{Q}\_{\mathbf{e},T}^{(-)}\mathbf{S}\_{\mathbf{a}}(t) = e^{i\alpha\_{\mathbf{e}}(T-t)}\mathbf{P}\_{\mathbf{S}\_{\mathbf{a}}}^{\mathbf{e}}(T-t) \tag{40}$$

In Eqs. (39) and (40) P*<sup>e</sup> Sa* ð Þ *T* � *t* are simple projection operators [11]. Denoting the Fourier transform like

$$\bigcirc \mathsf{F}[f(t)] \equiv \mathsf{F}(w) \tag{41}$$

Then, the Fourier transform of the GPO is

$$\mathbb{E}\_{\mathsf{C}}\mathsf{F}\Big[\mathsf{P}\_{\mathsf{S}\_{a}}^{\mathsf{c}}(\mathsf{t})\mathsf{e}^{i\alpha\_{\mathsf{t}}\mathsf{t}}\Big] = \sum\_{n=-\infty}^{\infty} \mathsf{C}\_{n,\mathsf{c}}|\mathsf{2}a\_{\mathsf{c}}|p\_{4\mathsf{z}a\_{\mathsf{c}}}(\mathsf{2}(\mathsf{w}-\mathsf{o}\_{\mathsf{c}})a\_{\mathsf{c}})\mathsf{e}^{-i(\mathsf{o}-\mathsf{o}\_{\mathsf{c}})\mathsf{z}\mathsf{m}} \equiv F\_{\mathsf{S}\_{a}}(\mathsf{w}-\mathsf{o}\_{\mathsf{c}}) \tag{42}$$

Where *p*4*πω<sup>e</sup>* ð Þ 2ð Þ *ω* � *ω<sup>e</sup> ω<sup>e</sup>* is a rectangular function. And for the convolution we have

$$\mathbb{E}\_{\mathbb{S}}\mathfrak{p}^{\varepsilon}\Big[\mathbf{P}\_{\mathcal{S}\_{a}}^{\varepsilon}(t)\mathfrak{e}^{ia\_{\mathbb{H}}t}\*\mathbf{P}\_{\mathcal{S}\_{b}}^{\varepsilon}(t)\mathfrak{e}^{ia\_{\mathbb{H}}t}\Big] = F\_{\mathcal{S}\_{a}}(\boldsymbol{\alpha}-\boldsymbol{\alpha}\_{\varepsilon})F\_{\mathcal{S}\_{b}}(\boldsymbol{\alpha}-\boldsymbol{\alpha}\_{\mathbb{H}})\tag{43}$$

Then we see that the set of Fourier transforms of the GPOs behaves like a set of orthogonal basis functions for the frequency domain, that is, the resonant functions *ym <sup>e</sup>* ð Þ **r**;*ω* as we can verify in **Figure 1**. So the GPO can be considered as the forerunners of the Zap projection operators.

## **9. Conclusions**

We can conclude that the Zap projection operators (ZPO) can be used as an alternative approach to the generalized projection operators (GPO) that is like an alternative for clean the evanescent signals [10] from disturbances generated by the sources and at the same time to clean the source signals from resonant solutions. We can also use the two classes of projectors in a consecutively manner. The former vision suppose that the evanescent waves [10] can be considered as part of the conventional traveling waves like an everything and that we must take away the effect of the resonances with the application of the *<sup>Q</sup>* operator. In any case we have shown the power of the Fourier transform applied to mathematical analysis in broadcasting problems and to physically characterize and solve them.

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