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

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