A stability problem for some complete and minimal Gabor systems in $L^2(\mathbb{R})$

Abstract

A Gabor system in $L^2(\mathbb{R})$, generated by a window $g\in L^2(\mathbb{R})$ and associated with a sequence of times and frequencies $\Gamma\subset\mathbb{R}^2$, is a set formed by translations in time and modulations of $g$. In this paper we consider the case when $g$ is the Gaussian function and $\Gamma$ is a sequence whose associated Gabor system $\mathcal{G}_\Gamma$ is complete and minimal in $L^2(\mathbb{R})$. We consider two main cases: that of the lattice without one point and that of the sequence constructed by Ascensi, Lyubarskii and Seip lying on the union of the coordinate axes of the time-frequency space. We study the stability problem for these two systems. More precisely, we describe the perturbations of $\Gamma$ such that the associated Gabor systems remain to be complete and minimal. Our method of proof is based essentially on estimates of some infinite products.