An $L^2-$stability estimate for periodic nonuniform sampling in higher dimensions

TL;DR

This paper develops an $L^2$-stability estimate for reconstructing multivariate bandlimited functions with disjoint spectra from periodic nonuniform samples, using Vandermonde matrices to ensure unique recovery.

Contribution

It introduces explicit $L^2$-stability estimates for reconstructing multivariate bandlimited functions from periodic nonuniform samples, leveraging Vandermonde matrix invertibility.

Findings

Explicit formulas relating functions to samples for disjoint spectral bands

Invertibility of Vandermonde matrices guarantees unique recovery

Provides $L^2$-stability estimates for reconstruction

Abstract

We consider sampling strategies for a class of multivariate bandlimited functions $f$ that have a spectrum consisting of disjoint frequency bands. Taking advantage of the special spectral structure, we provide formulas relating $f$ to the samples $f(y), y\in X$, where $X$ is a periodic nonuniform sampling set. In this case, we show that the reconstruction can be viewed as an iterative process involving certain Vandermonde matrices, resulting in a link between the invertibility of these matrices to the existence of certain sampling sets that guarantee a unique recovery. Furthermore, estimates of inverse Vandermonde matrices are used to provide explicit $L^{2}$-stability estimates for the reconstruction of this class of functions.