Tag - Fourier analysis

Maryna Viazovska: Fourier Uniqueness and Interpolation

25th October, 2023

Can we reconstruct a function by knowing only a subset of its values and a subset of the values of the function's Fourier transform?

How many values do we need to know for such a reconstruction? Can we interpolate a given subset of values? What are the possible applications of such interpolation? In this series of lectures, we will try to answer these questions.

In the first lecture, we will speak about the Cohn-Elkies linear programming bound for the sphere packing and how this bound's analysis led to the discovery of a Fourier interpolation formula. The second lecture will discuss explicit constructions of Fourier uniqueness sets and Fourier interpolation formulas. The third lecture will focus on analytic approaches to Fourier uniqueness and interpolation.

Mikhail Sodin: Fourier Uniqueness and Nonuniqueness Pairs

16th October, 2023

Motivated by a discovery by Radchenko and Viazovska and by a work by Ramos and Sousa, we find conditions sufficient for a pair of discrete subsets of the real axis to be a uniqueness or a non-uniqueness pair for the Fourier transform. These conditions are not too far from each other. The uniqueness theorem can be upgraded to the frame bound and an interpolation formula, which in turn produce an abundance of Poisson-like formulas (a.k.a. 'crystalline measures').