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

TL;DR

This paper investigates the stability of certain complete and minimal Gabor systems in $L^2(R)$ generated by Gaussian windows, analyzing how small perturbations affect their completeness and minimality, with methods based on infinite product estimates.

Contribution

It provides a detailed analysis of the stability of specific Gabor systems with Gaussian windows, focusing on perturbations that preserve their completeness and minimality.

Findings

Identifies conditions under which Gabor systems remain complete and minimal after perturbations.

Analyzes two main cases: lattice missing one point and sequences on coordinate axes.

Uses estimates of infinite products to establish stability results.

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.