Fourier uniqueness and non-uniqueness pairs

TL;DR

This paper establishes conditions under which pairs of discrete real line subsets are unique or non-unique for the Fourier transform, and extends the results to interpolation formulas and measures.

Contribution

It provides new sufficient conditions for Fourier uniqueness and non-uniqueness pairs, advancing understanding of Fourier transform pairs and their applications.

Findings

Identified conditions for Fourier uniqueness pairs

Derived interpolation formulas from the uniqueness results

Constructed discrete measures with discrete Fourier transforms

Abstract

Motivated by recent works by Radchenko and Viazovska and by Ramos and Sousa, we find sufficient conditions for a pair of discrete subsets of the real line to be a uniqueness or a non-uniqueness pair for the Fourier transform. These conditions are close to each other. The uniqueness result can be upgraded to an interpolation formula, which in turn produces an abundance of discrete measures with discrete Fourier transform.