I will begin with a beautiful and surprising Fourier interpolation formula discovered
by Danilo Radchenko and Marina Viazovska. Their formula enables the reconstruction of
a function from its values and the values of its Fourier transform on a particular
sequence of points. It yields what is probably the first examples of discrete Fourier
uniqueness pairs. Their work relies on the theory of modular forms.
I will then present a purely analytic approach to discrete Fourier uniqueness and
non-uniqueness pairs, as well as to interpolation formulas, in which rigid arithmetic
constraints on the nodes are replaced by density conditions. This approach applies
in both one and several dimensions.
The talk is based on ongoing work with Fedor Nazarov and Aleksei Kulikov (part of
which can be found in arXiv:2306.14013)