On the time-frequency representation of operators and generalized Gabor multiplier approximations

The problem of representation and approximation of linear operators by means of modification in the time-frequency domain is considered. Before turning to the discrete and sub-sampled case, a complete characterization of linear operators by means of a twisted convolution in the continuous time-frequency domain is suggested. Subsequently, existing results on approximation by time-frequency multipliers are reviewed. To overcome the limitations imposed by these multipliers, two more general constructions are proposed, termed multiple Gabor multipliers and Twisted Spline type functions. Conditions ensuring the existence of optimal multiple Gabor multipliers are given. As the constructions suggested in this paper are mainly based on the Weyl-Heisenberg group, twisted convolution plays a central role in the results described in this paper.