Some general preliminary considerations

1. Each function $g(x)\in\EuScript{L}^1(\Omega)$ defines a functional $\varphi\in\DSp$ in the following way: $$ \varphi(f)=\int\limits_{\Omega} g(x)\,f(x)\,d\mu. $$ Such functional we will call regular functional and function $g$ -- its generator.
2. For each $x\in\Omega$ we will consider a functional $\delta_x\in\DSp$ defined as follows: \begin{equation}\label{dFn} \delta_x(f)=f(x). \end{equation} Since generally, we can not speak about values at the point for functions from $\EuScript(L)^\infty$ in the following, we assume some regularity for functions from considered spaces, so that () is correctly defined.

Necessary notations and motivation

Consider an operator equation: \begin{equation}\label{OpEq} Af=g \end{equation}where $f\in\FOx$ is unknown and $g\in\GOy$ is given. We are interested to have an integral representation for solution of (). For this purpose we write: $$ f(x)=\delta_x(f)=\delta_x(A^{-1}(g))=[\, (A^{-1})^*\delta_x \,](g). $$

Definition of Green's function

<</SPAN>#57#>Green's function of operator $A$ and solution of () admits the following integral representation: $$ f(x)=\int\limits_{\Omega_y}G(x,y)\,g(y)\,d\mu_y $$