Multivariate Gabor frames for operators in matrix-valued signal spaces over locally compact abelian groups

Abstract

In this paper, we study multivariate Gabor frames in matrix-valued signal spaces over locally compact abelian (LCA) groups, where the lower frame condition depends on a bounded linear operator [Formula: see text] on the underlying matrix-valued signal space. This type of Gabor frame is also known as a multivariate [Formula: see text]-Gabor frame. By extending work of G[Formula: see text]vruta, we present necessary and sufficient conditions for the existence of [Formula: see text]-Gabor frames of multivariate matrix-valued Gabor systems. Some operators which can transform multivariate matrix-valued Gabor and [Formula: see text]-Gabor frames into [Formula: see text]-Gabor frames in terms of adjointable operators are discussed. Finally, we give a Paley–Wiener-type perturbation result for multivariate matrix-valued [Formula: see text]-Gabor frames.