Integral Operators Basic in Random Fields Estimation Theory

TL;DR

This paper analyzes the properties of a weakly singular integral operator arising in random field estimation, proving it is an isomorphism between Sobolev spaces via reduction to an elliptic boundary value problem.

Contribution

It establishes the isomorphism of the integral operator in Sobolev spaces by linking it to an elliptic boundary value problem, addressing complexities of negative order Sobolev spaces.

Findings

The integral operator is weakly singular and of the first kind.

It is an isomorphism between appropriate Sobolev spaces.

The proof involves reduction to an elliptic boundary value problem.

Abstract

The paper deals with the basic integral equation of random field estimation theory by the criterion of minimum of variance of the error estimate. This integral equation is of the first kind. The corresponding integra$ operator over a bounded domain $\Omega $ in ${\Bbb R}^{n}$ is weakly singular. This operator is an isomorphism between appropriate Sobolev spaces. This is proved by a reduction of the integral equ$ an elliptic boundary value problem in the domain exterior to $\Omega .$ Extra difficulties arise due to the fact that the exterior boundary value problem should be solved in the Sobolev spaces of negative order.