Signal analytic proofs of two basic results on Lattice expansions

We present new and short proofs of two theorems in the theory of lattice expansions. These proofs are based on a necessary and sufficient condition, found by Wexler and Raz, for biorthogonality. The first theorem is the Lyubarskii-Seip-Wallstén theorem for lattices, according to which the set of Gaussians 2^1/4 exp(-p(t - na)² + 2pimbt), n, m ¿ Z, constitutes a frame when a > 0,b > 0,ab < 1. In addition, we display dual functions for this case. The second theorem is the result that a set g_na,mb(t) = g(t - na) exp(2pimbt), n, m ¿ Z of time-frequency translates of a g ¿ L²(R) cannot be a frame when a > 0,b > 0,ab > 1.