Lattice tiling and density conditions for subspace Gabor frames

The well known density theorem in time-frequency analysis establishes the connection between the existence of a Gabor frame G(g, A, B) = {e(2 pi i < Bm,x >) g(x - An): m, n is an element of Z(d)} for L-2(R-d) and the density of the time-frequency lattice AZ(d) x BZ(d). This is also tightly related to lattice tiling and packing. In this paper we investigate the density theorem for Gabor systems in L-2(S) with S being an AZ(d)-periodic subset of R-d. We characterize the existence of a Gabor frame for L-2(S) in terms of a condition that involves the Haar measure of the group generated by AZ(d) and (B-t)(-1)Z(d). This new characterization is used to recover the density theorem and several related known results in the literature. Additionally we apply this approach to obtain the density theorems for multi-windowed and super Gabor frames for L-2(S). (C) 2013 Elsevier Inc. All rights reserved.