Discrete Sampling and Interpolation: Universal Sampling Sets for Discrete Bandlimited Spaces

TL;DR

This paper investigates universal sampling sets for discrete signals in bandlimited spaces, providing characterizations, structure theorems, algorithms for construction, and applications to uncertainty principles, especially when the signal length is a prime power.

Contribution

It introduces new characterizations and algorithms for constructing universal sampling sets in discrete bandlimited spaces, with a focus on prime power lengths.

Findings

Characterizations of universal sampling sets for prime power lengths

Algorithms for constructing universal sampling sets

Applications to additive uncertainty principles

Abstract

We study the problem of interpolating all values of a discrete signal f of length N when d<N values are known, especially in the case when the Fourier transform of the signal is zero outside some prescribed index set J; these comprise the (generalized) bandlimited spaces B^J. The sampling pattern for f is specified by an index set I, and is said to be a universal sampling set if samples in the locations I can be used to interpolate signals from B^J for any J. When N is a prime power we give several characterizations of universal sampling sets, some structure theorems for such sets, an algorithm for their construction, and a formula that counts them. There are also natural applications to additive uncertainty principles.