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 $\Omega$ (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 $\Omega$.

17 pages

References in corpus (2)

• A sharp upper bound for sampling numbers in $L_2$
• Random Sampling in reproducing kernel subspaces of $L^p({\mathbb{R}}^n)$