Random sampling in reproducing kernel spaces with mixed norm

Abstract

In this work we consider random sampling of signals in (in)finite-dimensional reproducing kernel spaces with mixed norm. Here the random sampling refers to randomly taken sampling positions according to some probability measure. We study the stability of random sampling procedure by establishing sampling inequality that holds with high probability when the sampling size is large. We establish the probabilistic sampling inequality though a combination of mathematical analysis and probabilistic analysis. The main tools we use are covering number of signal (function) space and (uniform) large deviation inequality for a sequence of random variables. We provide a concise proof and our proof leads to explicit and transparent estimates involved in the probability with which the sampling inequality holds.