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Transactions of the American Mathematical Society

Published by the American Mathematical Society, the Transactions of the American Mathematical Society (TRAN) is devoted to research articles of the highest quality in all areas of pure and applied mathematics.

ISSN 1088-6850 (online) ISSN 0002-9947 (print)

The 2020 MCQ for Transactions of the American Mathematical Society is 1.43.

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Dilation of the Weyl symbol and Balian-Low theorem
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by Gerard Ascensi, Hans G. Feichtinger and Norbert Kaiblinger PDF
Trans. Amer. Math. Soc. 366 (2014), 3865-3880 Request permission

Abstract:

The key result of this paper describes the fact that for an important class of pseudodifferential operators the property of invertibility is preserved under minor dilations of their Weyl symbols. This observation has two implications in time-frequency analysis. First, it implies the stability of general Gabor frames under small dilations of the time-frequency set, previously known only for the case where the time-frequency set is a lattice. Secondly, it allows us to derive a new Balian-Low theorem (BLT) for Gabor systems with window in the standard window class and with general time-frequency families. In contrast to the classical versions of BLT the new BLT does not only exclude orthonormal bases and Riesz bases at critical density, but indeed it even excludes irregular Gabor frames at critical density.
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Additional Information
  • Gerard Ascensi
  • Affiliation: Departament de Matemàtica Aplicada i Anàlisi, Universitat de Barcelona, Gran Via 585, 08007 Barcelona, Spain
  • Email: gerard.ascensi@ub.edu
  • Hans G. Feichtinger
  • Affiliation: Faculty of Mathematics, University of Vienna, Nordbergstraße 15, 1090 Vienna, Austria
  • MR Author ID: 65680
  • ORCID: 0000-0002-9927-0742
  • Email: hans.feichtinger@univie.ac.at
  • Norbert Kaiblinger
  • Affiliation: Institute of Mathematics, University of Natural Resources and Life Sciences Vienna, Gregor-Mendel-Strasse 33, 1180 Vienna, Austria
  • Email: norbert.kaiblinger@boku.ac.at
  • Received by editor(s): May 3, 2011
  • Received by editor(s) in revised form: November 30, 2012
  • Published electronically: December 6, 2013
  • Additional Notes:

    The authors were supported by the Austrian Science Fund FWF grants M1149

    (first author), P20442

    (second author), and P21339

    , P24828

    (third author).

  • © Copyright 2013 American Mathematical Society
    The copyright for this article reverts to public domain 28 years after publication.
  • Journal: Trans. Amer. Math. Soc. 366 (2014), 3865-3880
  • MSC (2010): Primary 47G30; Secondary 42C15, 81S30
  • DOI: https://doi.org/10.1090/S0002-9947-2013-06074-6
  • MathSciNet review: 3192621