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Representation of Operators in the Time-Frequency Domain and Generalzed Gabor Multipliers

Authors: Monika Dörfler and Bruno Torrésani

Published in: Dagstuhl Seminar Proceedings, Volume 8492, Structured Decompositions and Efficient Algorithms (2009)

Abstract

Starting from a general operator representation in the time-frequency domain, this paper addresses the problem of approximating linear operators by operators that are diagonal or band-diagonal with respect to Gabor frames. A haracterization of operators that can be realized as Gabor multipliers is given and necessary conditions for the existence of (Hilbert-Schmidt) optimal Gabor multiplier approximations are discussed and an efficient method for the calculation of an operator's best approximation by a Gabor multiplier is derived. The spreading function of Gabor multipliers yields new error estimates for these approximations. Generalizations (multiple Gabor multipliers) are introduced for better approximation of overspread operators. The Riesz property of the projection operators involved in generalized Gabor multipliers is characterized, and a method for obtaining an operator's best approximation by a multiple Gabor multiplier is suggested. Finally, it is shown that in certain situations, generalized Gabor multipliers reduce to a finite sum of regular Gabor multipliers with adapted windows.

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Monika Dörfler and Bruno Torrésani. Representation of Operators in the Time-Frequency Domain and Generalzed Gabor Multipliers. In Structured Decompositions and Efficient Algorithms. Dagstuhl Seminar Proceedings, Volume 8492, pp. 1-28, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2009)

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@InProceedings{dorfler_et_al:DagSemProc.08492.7,
  author =	{D\"{o}rfler, Monika and Torr\'{e}sani, Bruno},
  title =	{{Representation of Operators in the Time-Frequency Domain and Generalzed Gabor Multipliers}},
  booktitle =	{Structured Decompositions and Efficient Algorithms},
  pages =	{1--28},
  series =	{Dagstuhl Seminar Proceedings (DagSemProc)},
  ISSN =	{1862-4405},
  year =	{2009},
  volume =	{8492},
  editor =	{Stephan Dahlke and Ingrid Daubechies and Michal Elad and Gitta Kutyniok and Gerd Teschke},
  publisher =	{Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
  address =	{Dagstuhl, Germany},
  URL =		{https://drops.dagstuhl.de/entities/document/10.4230/DagSemProc.08492.7},
  URN =		{urn:nbn:de:0030-drops-18808},
  doi =		{10.4230/DagSemProc.08492.7},
  annote =	{Keywords: Operator approximation, generalized Gabor multipliers, spreading function, twisted convolution}
}

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Representation of Operators in the Time-Frequency Domain and Generalzed Gabor Multipliers

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