Jorge Antezana: Fourier Frames near the critical density in LCA groups

Abstract: 
Gröchenig, Kutyniok and Seip generalized the notions of upper and lower densities, originally introduced by  Beurling in \(R^d\), to locally compact abelian (LCA) groups. Roughly speaking, these densities measure how a set of characters is distributed with respect to certain canonical reference lattice in the dual group.

Given a relatively compact set \(\Omega\) in a LCA group G,  Gröchenig et. al. proved that a set of characters \(\Gamma\) is a frame for \(L^2(\Omega)\) only if the density of \(\Gamma\) is greater or equal to the Haar measure of \(\Omega\). Here, the Haar measure is normalized conveniently with respect to the aforementioned reference lattice. However, the existence of a set of characters with density arbitrary close to the Haar measure of \(\Omega\) remained open.  

In this talk we will discuss the solution to this problem, which is a joint work with Elona Agora and Carlos Cabrelli.