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)={e2πi〈Bm,x〉g(x−An):m,n∈Zd}G(g,A,B)={e2πi〈Bm,x〉g(x−An):m,n∈Zd} for L2(Rd)L2(Rd) and the density of the time-frequency lattice AZd×BZdAZd×BZd. This is also tightly related to lattice tiling and packing. In this paper we investigate the density theorem for Gabor systems in L2(S)L2(S) with S   being an AZdAZd-periodic subset of RdRd. We characterize the existence of a Gabor frame for L2(S)L2(S) in terms of a condition that involves the Haar measure of the group generated by AZdAZd and (Bt)−1Zd(Bt)−1Zd. 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 L2(S)L2(S).