Probabilistic regularization of Fredholm integral equations of the first kind

Abstract

The main purpose of this paper is to focus on various issues inherent to the regularization theory of Fredholm integral equations of the first kind. Particular attention is devoted to the probabilistic approach to regularization, and a regularizing algorithm based on statistical methods is then proposed and tested on examples. The information theory approach is studied from two different viewpoints: the first approach is the standard one based on probability theory; the second one is formulated, in analogy with communication theory, in terms of the $\varepsilon $-capacity in the sense elaborated by Kolmogorov and his school. The classical problem of the resolving power in optics is then used to exemplify the relation between these two approaches.