On matrix-valued Gabor frames over locally compact abelian groups

Abstract

In this paper, we study Gabor frames in the matrix-valued signal space [Formula: see text], where [Formula: see text] is a locally compact abelian group which is metrizable and [Formula: see text]-compact, and [Formula: see text] is a positive integer. First, we give sufficient conditions on scalars in an infinite combination of vectors (from a given matrix-valued Gabor frame) to constitute a new frame for the space [Formula: see text]. This generalizes a result due to Aldroubi. Second, we discuss frame conditions for finite sums of matrix-valued Gabor frames. Sufficient conditions for finite sums of matrix-valued Gabor frames in terms of frame bounds are established. It is shown that the sum of images of matrix-valued Gabor frames under bounded linear operators acting on [Formula: see text] constitute a frame for the space [Formula: see text] provided operators are adjointable with respect to the matrix-valued inner product and satisfy a majorization. Finally, we show that matrix-valued Gabor frames are stable under small perturbations.