Sampling properties of zeros of Gaussian entire functions

Felipe Marceca, Jeremiah Buckley, Joaquin Singer

Session: Frames, Riesz bases, and related topics (Jorge Antezana)

Keywords: Fock spaces, sampling, Gaussian analytic functions

TL;DR: Sampling properties of zeros of Gaussian entire functions

Abstract:

In this talk we study sampling properties of zeros of Gaussian entire functions on Fock spaces. Sampling sets on Fock spaces were characterized by Seip and Wallstén in terms of their density and separation properties. These requirements are global and therefore too strict for many random point processes of interest that usually exhibit low correlation at large distances. We prove random sampling inequalities for polynomials of degree $\le d$ using ${d}+o(d)$ points, and show that the sampling constants grow slower than $d^\varepsilon$ for any $\varepsilon>0$.
In particular, we recover a result from Lyons and Zhai in the case of Gaussian entire functions, where it is shown that their zeros are uniqueness sets on the Fock space.
Joint work with Jerry Buckley and Joaquín Singer.

Submission Number: 42