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

TL;DR

This paper investigates the stability of random sampling in localized reproducing kernel spaces, showing that a logarithmic number of random points can reliably sample functions concentrated on a compact set.

Contribution

It establishes probabilistic stability results for random sampling sets in localized reproducing kernel spaces, linking sampling density to measure and error bounds.

Findings

Logarithmic number of random points ensures stable sampling with high probability.

Sampling stability depends on discretizing the integral norm of simple functions.

Results apply to functions concentrated on compact sets in $L^p$ spaces.

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 $\Omega$ (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 $\Omega$. Moreover, we prove with an overwhelming probability that ${\mathcal O}(\mu(\Omega)(\log \mu(\Omega))^3)$ many random points uniformly distributed over $\Omega$ yield a stable set of sampling for functions concentrated on $\Omega$.