Gabor multipliers with varying lattices

In the early days of Gabor analysis it was a common to say that Gabor expansions of signals are interesting due to the natural interpretation of Gabor coefficients, but unfortunately the computation of Gabor coefficients is costly. Nowadays a large variety of efficient numerical algorithms exists (see [28] for a survey) and it has been recognized that stable and robust Gabor expansions can be achieved at low redundancy, e.g., by using a Gaussian atom and any time-frequency lattice of the form aZ×bZ C IR with ab < 1. Consequently Gabor multipliers, i.e., linear operators obtained by applying a pointwise multiplication of the Gabor coefficients, become an important class of time-variant filters. It is the purpose of this paper to describe the fact that provided one uses Gabor atoms from a suitable subspace S0(R) ⊆ L(R) one has the expected continuous dependence of Gabor multipliers on the ingredients. In particular, we will provide new results which show that a small change of lattice parameters implies only a small change of the corresponding Gabor multiplier (e.g., in the Hilbert-Schmidt norm).