Fourier uniqueness and non-uniqueness pairs

12.05.2023 09:45 - 11:15

Aleksei Kulikov (Tel Aviv University)

Given discrete sets A, B in R, we call then we call them a Fourier uniqueness pair if there does not exist a non-trivial Schwartz function f such that f is zero on A and its Fourier transform is zero on B. In their breakthrough paper, Radchenko and Viazovska constructed first such example with A = B = {±n1/2, n in N} and moreover, they provided a way to reconstruct a function from the values of it and its Fourier transform on this set. Later, Ramos and Sousa explored more general Fourier uniqueness sets and they showed, in particular, that if t < 1-√2 /2 then A=B={±nt, n in N} is a Fourier uniqueness pair.    

Motivated by these results, we studied necessary and sufficient conditions for a pair of sets to be a Fourier uniqueness pair. We showed that if A=B={±an1/2,  n in N} then (A, B) is a uniqueness pair if a < 1 and a non-uniqueness pair if a > 1, and more generally we classified all polynomial uniqueness and non-uniqueness pairs up to the endpoint. Moreover,  in the uniqueness case the result can be improved to the frame bound and consequentially the interpolation formula similar to the one constructed by Radchenko and Viazovska.  This in turn can be used to construct an abundance of new crystalline measures.    

The talk is based on a joint work with Fedor Nazarov and Mikhail Sodin.


K. Gröchenig, L. Liehr, J. L. Romero and I. Shafkulovska