Random Sampling in Reproducing Kernel Subspace of Mixed Lebesgue Spaces

Abstract

In this article, we consider the random sampling in the image space $V$ of mixed Lebesgue space $L^{p,q}(\mathbb{R}^{n+1})$ under an idempotent integral operator. We assume some decay and regularity conditions of the kernel and approximate the unit sphere in $V$ on a bounded cube $C_{R,S}$ by a finite-dimensional subspace of $V$. Consequently, the set of concentrated functions is totally bounded. We prove with an overwhelming probability that the random sample set uniformly distributed over $C_{R,S}$ is a stable set of sampling for the set of concentrated functions on $C_{R,S}$. Moreover, we propose an iterative scheme to reconstruct the concentrated signal from its random measurements.