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Starting from a general operator representation in the time-frequency domain,this paper addresses the problem of approximating linear operators by operatorsthat are diagonal or band-diagonal with respect to Gabor frames. Acharacterization of operators that can be realized as Gabor multipliers isgiven and necessary conditions for the existence of (Hilbert-Schmidt) optimalGabor multiplier approximations are discussed and an efficient method for thecalculation of an operator's best approximation by a Gabor multiplier isderived. The spreading function of Gabor multipliers yields new error estimatesfor these approximations. Generalizations (multiple Gabor multipliers) areintroduced for better approximation of overspread operators. The Riesz propertyof the projection operators involved in generalized Gabor multipliers ischaracterized, and a method for obtaining an operator's best approximation by amultiple Gabor multiplier is suggested. Finally, it is shown that in certainsituations, generalized Gabor multipliers reduce to a finite sum of regularGabor multipliers with adapted windows.

Representation of operators in the time-frequency domain and generalized Gabor multipliers