Exponential Approximation of Band-limited Functions from Nonuniform Sampling by Regularization Methods

TL;DR

This paper demonstrates that band-limited functions can be reconstructed from nonuniform samples with exponential accuracy using Gaussian regularization, extending known results from uniform sampling.

Contribution

It proves that exponential convergence in reconstruction is achievable for nonuniform sampling via Gaussian regularization, a previously unresolved question.

Findings

Exponential convergence achieved with Gaussian regularized nonuniform sampling series.

Constructive methods provided for recovering band-limited functions.

Several examples illustrating exponential convergence are presented.

Abstract

Reconstructing a band-limited function from its finite sample data is a fundamental task in signal analysis. A Gaussian regularized Shannon sampling series has been proved to be able to achieve exponential convergence for uniform sampling. Whether such an exponential convergence can also be achieved for nonuniform sampling by regularization methods was unresolved. In this paper, we give an affirmative and constructive answer to this question. Specifically, we show that one can recover a band-limited function by Gaussian or hyper-Gaussian regularized nonuniform sampling series with an exponential convergence rate. Our analysis is based on the residue theorem in complex analysis, which is used to represent the truncated error by a contour integral. Several concrete examples of nonuniform sampling with exponential convergence will be presented.