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Gabor Schauder bases and the Balian-Low theorem

J. Math. Phys. 47, 113506 (2006); doi:10.1063/1.2360041

Published 21 November 2006

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Christopher Heil
School of Mathematics, Georgia Institute of Technology, Atlanta, Georgia 30332-0160

Alexander M. Powell
Department of Mathematics, Vanderbilt University, Nashville, Tennessee 37240
The Balian-Low Theorem is a strong form of the uncertainty principle for Gabor systems that form orthonormal or Riesz bases for L2([openface R]). In this paper we investigate the Balian-Low Theorem in the setting of Schauder bases. We prove that new weak versions of the Balian-Low Theorem hold for Gabor Schauder bases, but we constructively demonstrate that several variants of the BLT can fail for Gabor Schauder bases that are not Riesz bases. We characterize a class of Gabor Schauder bases in terms of the Zak transform and product [script A]2 weights; the Riesz bases correspond to the special case of weights that are bounded away from zero and infinity. ©2006 American Institute of Physics
History: Received 23 August 2006; accepted 13 September 2006; published 21 November 2006
Permalink: http://link.aip.org/link/?JMAPAQ/47/113506/1
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KEYWORDS and PACS

Keywords
PACS
  • 03.65.Ta
    Foundations of quantum mechanics; measurement theory
  • 02.30.Uu
    Integral transforms
  • YEAR: 2006

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ISSN:
0022-2488 (print)   1089-7658 (online)
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