Representation of Operators in the Time-Frequency Domain and Generalized Gabor Multipliers |
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Authors: | Monika Dörfler Bruno Torrésani |
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Institution: | 1. Numerical Harmonic Analysis Group, Faculty of Mathematics, University of Vienna, Alserbachstra?e 23, 1090, Wien, Austria 2. Laboratoire d’Analyse, Topologie et Probabilités, Centre de Mathématique et d’Informatique, 39 rue Joliot-Curie, 13453, Marseille cedex 13, France
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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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