Density of Gabor frames
type: Item
instance of: scholarly article
title: Density of Gabor frames
zbMATH Open document ID: 0960.42007
author: Ole Christensen
author: Baiqiao Deng
author: Christopher Heil
published in: Applied and Computational Harmonic Analysis
publication date: 2000-08-20T00:00:00Z
review text: A Gabor system is a system of time-frequency shifts \(S(g,\Lambda)=\{ e^{2\pi i bx}g(x-a){\}}_{(a,b)\in\Lambda}\) of a function \(g\in L^2(R)\) and a set of time-frequency shifts \(\Lambda\subset R^{2d}\). The authors obtain necessary conditions for a finite union \(S=\bigcup_{k=1}^rS(g_k,\Lambda_k)\) to be frame or Riesz basis, in terms of Beurling densities of \(\Lambda=\bigcup_{k=1}^r\Lambda_k\). They show (Theorem 1.1 and Corollary 3.7): (i) if \(S\) is a frame then \(1\leq D^{-}(\Lambda)\leq D^{+}(\Lambda)<\infty\); (ii) if \(S\) is a Riesz basis for \(L^2(R^d)\), then \(D^{+}(\Lambda)=D^{-}(\Lambda)=1\). In particular, it is shown that no collection \(\bigcup_{k=1}^r\{g_k(x-a){\}}_{a\in\Gamma_k}\) of pure translates can form a frame for \(L^2(R^d)\) (Theorem 1.2).
reviewed by: Radu Balan
Mathematics Subject Classification ID: 42C40
zbMATH DE Number: 1384042
zbMATH Keywords: Beurling density
zbMATH Keywords: Gabor frame
zbMATH Keywords: Gabor Riesz basis
MaRDI profile type: Publication
full work available at URL: acha.1999.0271
OpenAlex ID: W1972151147
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DOI: 10.1006/ACHA.1999.0271
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description: scientific article; zbMATH DE number 1384042