A First Survey of Gabor Multipliers

H.G. Feichtinger

Department of Mathematics, University of Vienna, Strudlhofgasse 4, A-1090, Vienna, Austria
email: hans.feichtinger@univie.ac.at

K. Nowak

Department of Mathematics and Computer Science, Drexel University, 3141 Chestnut Street, Philadelphia, PA 19104-2875, USA
email: knowak@mcs.drexel.edu

We describe various basic facts about Gabor multipliers and their continuous analogue which we will call STFT-multipliers. These operators are obtained by going from the signal domain to some transform domain, and applying a pointwise multiplication operator before resynthesis. Although such operators have been in use implicitly for quite some time, this paper appears to be the first systematic mathematical treatment of Gabor multipliers. Indeed, typical time-frequency localization operators, or thresholding algorithms involve simple $0/1$-multipliers. The main results of this chapter are of a qualitative nature and describe how the properties of the Gabor multiplier depend on the decay of the multiplier sequence, the time-frequency (TF) concentration properties of the Gabor atom in use, and the time-frequency-lattice. These properties will be described in terms of the mapping properties of the corresponding Gabor multiplier between modulation spaces, or membership in some operator ideal (such as trace-class or Hilbert--Schmidt operator). It is also possible to give relatively precise estimates on behaviour of the sequence of eigenvalues of such operators, especially for the case of tight Gabor frames. We shall also discuss the problem of injectivity of the linear mapping from the multiplier symbol to the operator, recovery of Gabor multipliers from lower symbols, and a related question concerning best approximation of operators (e.g., from the Hilbert--Schmidt class) by Gabor multipliers of a certain type.