**3. Connection between the eigenvalues** *η<sup>e</sup>* **and the function***λ*

We know that because of the Hilbert-Schmidt theory [2, 3] and more recently by our previous results [1], the solutions of Eq. (3) that is all the *y<sup>m</sup> <sup>e</sup>* ð Þ **r**; *ω* , form a set of orthogonal functions and then a set of eigenvalues *ηe*ð Þ *ω* **.** Thus we can relate the functions appearing in Eqs. (1) and (3) as follows:

By means of the spectral representation of Green function, [2, 3] we have:

$$\mathbf{G}\_{m}^{n(\circ)}(w;\mathbf{r},\mathbf{s}) = \sum\_{\varepsilon} \mathbf{C}\_{\varepsilon} \frac{\mathbf{y}\_{\varepsilon}^{m}(\mathbf{r})\mathbf{y}\_{\varepsilon}^{n}(\mathbf{s})}{\lambda - \eta\_{\varepsilon}} \tag{4}$$

And also

solutions. Indeed, it seems to be natural that the new operators include the Green function and are close to the Fredholm operator [2, 3]. Before we define the ZIO operators we must underline the fact that in a broadcasting situation [8–10] we must take into accounts not only one kind of traveling waves but all the known ones because the complete description of the phenomena comes from the GIFE. Another important goal of this paper is to give an explanation of the simultaneous validity of two sets of boundary conditions that are very apart one to the other and the fact that there is a connection with other projection operators, the generalized projection operators (GPO) [11] that separates the constituents of a signal in orthogonal parts.

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

We remember that if we take the inhomogeneous vector integral Fredholm

**<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> (1)

*<sup>t</sup>* ð Þ *ω*; **r**, **s**

*<sup>n</sup>* ð Þ *ω*; **r**, **s** .

*<sup>n</sup>* **<sup>r</sup>**<sup>0</sup> ð Þ , *<sup>ω</sup> dr*<sup>0</sup> (2)

*<sup>e</sup> dr*<sup>0</sup> (3)

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

þ *λ ω*ð Þ

(may be a non-local potential) with the free Green function **G***<sup>t</sup>*ð Þ<sup>∘</sup>

þ *ν ω*ð Þ

*s*

Equation (GHFE) [1]:

**2**

*<sup>m</sup>*ð Þ¼ **<sup>r</sup>**,*<sup>ω</sup> <sup>s</sup>*

conventional behavior of the electromagnetic field [12].

*ym*

functions as we will see in the next sections.

∞ð

0

Where the kernel **<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, is the product of the interaction *Am*

value (j j *λ ω*ð Þ ≪ 1), we arrive to the integral equation we named the GIFE:

∞ð

0

As we have commented, Eq. (2) carries a mechanism that allows

*<sup>e</sup>* ð Þ¼ **r**; *ω ηe*ð Þ *ω*

This last equation is the one that have the property of represent a complete panorama in a broadcasting problem, that is describes both the resonant and the

simultaneously consider both types of solution. The so called generalized source is indeed a blend of integral operators as we will see with properties we want to visualize. But first we must present the Generalized Homogeneous Fredholm

∞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

Eq. (3) has a special index *e* that mean a specific resonance [1, 4, 5, 8, 9]. Among the three Eqs. (1), (2), and (3) there are a common ingredient, for each equation we have used different names: *λ*, *ν* and *η* [1–3] but any of them can be incorporated to the kernel or used as an independent function or even an eigenvalue. In order to connect the homogeneous and inhomogeneous equation we must define some

And we make the ansatz of two successive approximations (a second order approach) [9], by the consideration that *λ ω*ð Þ is a number with a very small absolute

*<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>*

**<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*<sup>s</sup>*

**2. Remembering the GIFE**

Eq. (1):

$$\begin{aligned} u^{\mathfrak{m}}(\mathbf{r}, \boldsymbol{\omega}) &= u^{\mathfrak{m}(\circ)}(\mathbf{r}, \boldsymbol{\omega}) \\ &- \sum\_{\epsilon=1}^{\infty} \left[ \frac{\mathsf{y}\_{\epsilon}^{\mathfrak{m}}(\mathbf{r}, \boldsymbol{\alpha}) \mathsf{y}\_{\epsilon}^{\mathfrak{n}}(\mathbf{r}', \boldsymbol{\alpha})}{\lambda(\boldsymbol{\alpha}) - \eta\_{\epsilon}(\boldsymbol{\alpha})} u^{\mathfrak{m}(\circ)}(\mathbf{r}'; \boldsymbol{\alpha}) d\boldsymbol{r}' \right] \end{aligned} \tag{5}$$

The orthogonality relation is

$$\int\_{0}^{\infty} \mathcal{y}\_{\epsilon}^{m}(\mathbf{r}; \boldsymbol{\alpha}) \mathcal{A}^{mn} \mathcal{y}\_{i}^{n}(\mathbf{r}; \boldsymbol{\alpha}) dr = 0 \text{ if } i \neq e \tag{6}$$
