Random Sampling in Reproducing Kernel Subspace of Mixed Lebesgue Spaces

TL;DR

This paper studies random sampling in the reproducing kernel subspace of mixed Lebesgue spaces, establishing stability of random samples and proposing an iterative reconstruction method for concentrated signals.

Contribution

It introduces a probabilistic framework for stable random sampling and reconstruction in mixed Lebesgue spaces under kernel decay and regularity conditions.

Findings

Random samples form a stable sampling set with high probability.

The set of concentrated functions is totally bounded.

An iterative scheme effectively reconstructs signals from random measurements.

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.