Discrete Gabor frames in $\ell ^2(\mathbb {Z}^d)$

Abstract:

The theory of Gabor frames for the infinite dimensional signal/ function space $L^{2}(\mathbb {R}^d)$ and for the finite dimensional signal space $\mathbb {R}^{d}$ (or $\mathbb {C}^{d}$) has been extensively investigated in the last twenty years. However, very little has been done for the Gabor theory in the infinite dimensional discrete signal space $\ell ^2(\mathbb {Z}^d)$, especially when $d > 1$. In this paper we investigate the general theory for discrete Gabor frames in $\ell ^2(\mathbb {Z}^d)$. We focus on a few fundamental aspects of the theory such as the density/incompleteness theorem for frames and super-frames, the characterizations for dual frame pairs and orthogonal (strongly disjoint) frames, and the existence theorem for the tight dual frame of the Gabor type. The existence result for Gabor frames (resp. super-frames) requires a generalization of a standard result on common subgroup coset representatives.