Gabor multipliers with varying lattices

Abstract

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 formwith 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-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).

Can any unconditionally convergent multiplier be transformed to have the symbol (1) and Bessel sequences by shifting weights?

“…Multipliers are interesting from a mathematical point of view. They have been investigated for Gabor frames [15,7,13], for fusion frames [1], for generalized frames [25] and p-frames in Banach spaces [24]. The concept of multipliers is naturally related to weighted frames [5,26] as well as matrix representation of operators [4].…”

Sobolev spaces of fractional order, Lipschitz spaces, readapted modulation spaces and their interrelations; applications

“…For an excellent, understandable, survey on amalgams on R and on groups G see Fournier and Stewart 1 [27]; the latter had introduced them in [28]. For a multivariate version see [29].…”