Signal analytic proofs of two basic results on lattice expansions

Abstract

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 21/4 exp(-π(t-na)2+2πimbt), 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 gna, mb(t) = g(t-na)exp(2πimbt), n, m∈ℤ of time-frequency translates of a g∈L2(ℝ) cannot be a frame when a>0, b>0, ab>1. © 1994 Academic Press, Inc.