Random Sampling in reproducing kernel subspaces of $L^p({\mathbb R}^n)$

TL;DR

This paper investigates random sampling in reproducing kernel subspaces of $L^p({ m R}^n)$, demonstrating approximation by finite-dimensional subspaces and probabilistic stability of uniform samples within a cube.

Contribution

It introduces new results on the approximation and stability of random samples in reproducing kernel subspaces under decay conditions of the kernel.

Findings

Any element in the subspace can be approximated by a finite-dimensional subspace.

Random points uniformly distributed over a cube form stable samples with high probability.

The results depend on decay conditions of the integral kernel.

Abstract

In this paper, we study random sampling on reproducing kernel space $V$, which is a range of an idempotent integral operator. Under certain decay condition on the integral kernel, we show that any element in $V$ can be approximated by an element in a finite-dimensional subspace of $V$. Moreover, we prove with overwhelming probability that random points uniformly distributed over a cube $C$ is stable sample for the set of functions concentrated on $C$