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Dilation of the Weyl symbol and Balian-Low theorem


Authors: Gerard Ascensi, Hans G. Feichtinger and Norbert Kaiblinger
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
Published electronically: December 6, 2013
MathSciNet review: 3192621
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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
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

DOI: https://doi.org/10.1090/S0002-9947-2013-06074-6
Keywords: Weyl symbol, pseudodifferential operator, dilation, Gabor frame, time-frequency set, Balian-Low theorem, sampling, interpolation, Bargmann-Fock space
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).
Article copyright: © Copyright 2013 American Mathematical Society
The copyright for this article reverts to public domain 28 years after publication.

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