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

Volume 254 / 2020

Youssef Omari Studia Mathematica 254 (2020), 45-75 MSC: Primary 30H20; Secondary 32A15, 30D20. DOI: 10.4064/sm190216-4-9 Published online: 6 March 2020

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 complete and minimal. Our method of proof is based essentially on estimates of some infinite products.

Authors

  • Youssef OmariLaboratory of Mathematical Analysis and Applications (LAMA)
    CeReMAR, Faculty of Sciences
    Mohammed V University in Rabat
    4 Av. Ibn Battouta, Rabat, Morocco
    e-mail

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