The Balian-Low theorem regularity of Gabor systems
type: Item
instance of: scholarly article
title: The Balian-Low theorem regularity of Gabor systems
zbMATH Open document ID: 1037.42030
DOI: 10.1007/BF02930696
author: John J. Benedetto
author: Wojciech Czaja
author: Przemysław Gadziński
author: Alexander M. Powell
published in: The Journal of Geometric Analysis
publication date: 2003-11-10T00:00:00Z
review text: The Gabor system generated by a function \(g\in L^2(\mathbb{R})\) is the collection of functions defined by \( \{e^{2\pi imx}g(x-n)\}_{m,n\in \mathbb{Z}}\). The well-known Balian-Low theorem states that if this Gabor system forms an orthonormal basis for \(L^2(\mathbb{R})\), then \( \int | xg(x)| ^2dx = \infty\) or \(\int | \gamma \hat{g}(\gamma)| ^2d\gamma =\infty\). Thus, a function \(g\) generating a Gabor orthonormal basis can not be well localized in both time and frequency. The main result of the paper proves, for any \(\varepsilon>0\) and \(A,B>0\) with \(1/A+1/B=1\), the existence of a Gabor orthonormal basis for which \( \int | g(x)| ^2\frac{1+ | x| ^A}{\log^{1+\varepsilon}(2+| x| )}dx <\infty\) and \(\int | \hat{g}(\gamma)| ^2\frac{1+ | \gamma| ^B}{\log^{2+\varepsilon}(2+| \gamma | )}d\gamma <\infty.\)
Mathematics Subject Classification ID: 42C15
Mathematics Subject Classification ID: 42C40
zbMATH DE Number: 1983429
zbMATH Keywords: Balian-Low theorem
zbMATH Keywords: Gabor system
zbMATH Keywords: orthonormal basis
MaRDI profile type: Publication
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description: scientific article; zbMATH DE number 1983429