Fredholm integral equations of the first kind and topological information theory

TL;DR

This paper explores regularization methods for solving ill-posed Fredholm integral equations of the first kind, focusing on eigenfunction expansions and $\

Contribution

It introduces a new perspective on truncation methods using $\\varepsilon$-coverings of compact sets for regularization.

Findings

Eigenfunction expansion methods can be regularized via truncation.

$\\varepsilon$-coverings provide a novel framework for analyzing truncation.

The approach offers insights into the stability of solutions.

Abstract

The Fredholm integral equations of the first kind are a classical example of ill-posed problem in the sense of Hadamard. If the integral operator is self-adjoint and admits a set of eigenfunctions, then a formal solution can be written in terms of eigenfunction expansions. One of the possible methods of regularization consists in truncating this formal expansion after restricting the class of admissible solutions through a-priori global bounds. In this paper we reconsider various possible methods of truncation from the viewpoint of the $\varepsilon$-coverings of compact sets.