Local Invertibility of Integral Operators with Analytic Kernels

TL;DR

This paper explores the conditions under which integral operators with analytic kernels are locally invertible, establishing a connection between local invertibility, kernel rank, and linear independence, with implications for inverse problems.

Contribution

It introduces the concept of local linear independence and links it to full rank almost everywhere, providing new criteria for local invertibility of integral operators with analytic kernels.

Findings

Local invertibility holds for kernels with locally linearly independent Taylor functions.

Full rank almost everywhere is equivalent to local linear independence.

Geodesic distance on a sphere is of full rank a.e., unlike Euclidean distance.

Abstract

The invertibility of integral linear operators is a major problem of both theoretical and practical importance. In this paper we investigate the relation between an operator invertibility and the rank of its integral kernel to develop a local inverse theory. An operator is called locally invertible provided that any function can be recovered from its transformed image if the latter is known in an arbitrary open subset of its domain, i.e., if its image is known locally. It turns out that the local invertibility holds for any analytical kernel whose Taylor functions are linearly independent in any open subset of their domain - the so-called local linear independence condition. We also establish an equivalence between local linear independence and the so-called full rank a.e. property. The latter can be described as follows: for any finite, random sample of points, the square matrix obtained by applying, pairwise, the kernel function on them, has full rank almost surely. As an illustration, we show that the geodesic distance function on a sphere in more than one dimensions is of full rank a.e., in contrast to the Euclidean distance which is not.