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

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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 characterization 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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Authors and Affiliations

  1. Numerical Harmonic Analysis Group, Faculty of Mathematics, University of Vienna, Alserbachstraße 23, 1090, Wien, Austria

    Monika Dörfler

  2. Laboratoire d’Analyse, Topologie et Probabilités, Centre de Mathématique et d’Informatique, 39 rue Joliot-Curie, 13453, Marseille cedex 13, France

    Bruno Torrésani

Authors
  1. Monika Dörfler
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  2. Bruno Torrésani
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Corresponding author

Correspondence to Monika Dörfler.

Additional information

Communicated by Karlheinz Gröchenig.

The first author has been supported by WWTF project MA07-025 and FWF grant T 384-N13.

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Dörfler, M., Torrésani, B. Representation of Operators in the Time-Frequency Domain and Generalized Gabor Multipliers. J Fourier Anal Appl 16, 261–293 (2010). https://doi.org/10.1007/s00041-009-9085-x

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  • DOI: https://doi.org/10.1007/s00041-009-9085-x

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