Random sampling of signals concentrated on compact set in localized reproducing kernel subspace of $L^p({\mathbb R}^n)$

Abstract

The paper is devoted to studying the stability of random sampling in a localized reproducing kernel space. We show that if the sampling set on Ω (compact) discretizes the integral norm of simple functions up to a given error, then the sampling set is stable for the set of functions concentrated on Ω. Moreover, we prove with an overwhelming probability that O(µ(Ω)(log µ(Ω)) 3 ) many random points uniformly distributed over Ω yield a stable set of sampling for functions concentrated on Ω.