Hampus Nyberg: Deep zero problems in the plane and the disc

Abstract:

A recently introduced type of problems are Deep zero problems which are concerned with large amounts of local information of analytic functions at a small number of points. Here, the analytic functions are restricted to lie in a reproducing kernel Hilbert space. We also restrict to functions on the plane and the disc. Our main interest will be in questions of uniqueness, but related interpolation and sampling issues will also be considered. In this talk we shall discuss some of the theory and development of Deep zero problems as well as some connections to other problems. In particular, we present a uniqueness result for the Bargmann-Fock space. That is, a function in the Bargmann-Fock space is uniquely determined if the information is distributed 1:2 or 1:3 between two points in an arithmetic fashion. This extends a previous result where the information is distributed 1 : 1.