Recovery of Missing Samples in Oversampling Formulas for Band Limited Functions

TL;DR

This paper investigates conditions under which missing samples in oversampling formulas for band-limited functions can be recovered, extending previous work on frames and shift-invariant spaces with practical numerical demonstrations.

Contribution

It provides a sufficient condition for recovering finite missing samples based on linear independence over trigonometric polynomial spaces, applicable to derivative sampling.

Findings

Established a linear independence condition for sample recovery.

Applied the theory to derivative sampling of any order.

Demonstrated the results with a numerical experiment.

Abstract

In a previous paper, the author constructed frames and oversampling formulas for band-limited functions, in the framework of the theory of shift-invariant spaces. In this article we study the problem of recovering missing samples. We find a sufficient condition for the recovery of a finite set of missing samples. The condition is expressed as a linear independence of the components of a vector W over the space of trigonometric polynomials determined by the frequencies of the missing samples. We apply the theory to the derivative sampling of any order and we illustrate our results with a numerical experiment.