Sampling and Reconstruction of Signals in a Reproducing Kernel Subspace of $L^p({\Bbb R}^d)$

TL;DR

This paper investigates stable sampling and reconstruction methods for signals in reproducing kernel subspaces of L^p(R^d), demonstrating convergence and error estimates for iterative algorithms under certain conditions.

Contribution

It introduces a framework for stable sampling and reconstruction in reproducing kernel subspaces of L^p(R^d), including convergence analysis of iterative algorithms.

Findings

Stable reconstruction from samples with small gap.

Exponential convergence of iterative algorithms.

Asymptotic pointwise error estimates.

Abstract

In this paper, we consider sampling and reconstruction of signals in a reproducing kernel subspace of $L^p(\Rd), 1\le p\le \infty$, associated with an idempotent integral operator whose kernel has certain off-diagonal decay and regularity. The space of $p$-integrable non-uniform splines and the shift-invariant spaces generated by finitely many localized functions are our model examples of such reproducing kernel subspaces of $L^p(\Rd)$. We show that a signal in such reproducing kernel subspaces can be reconstructed in a stable way from its samples taken on a relatively-separated set with sufficiently small gap. We also study the exponential convergence, consistency, and the asymptotic pointwise error estimate of the iterative approximation-projection algorithm and the iterative frame algorithm for reconstructing a signal in those reproducing kernel spaces from its samples with sufficiently small gap.