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–Wallsten 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 Z of time–frequency translates of a g ∈ L2( R ) cannot be a frame when a > 0,b > 0,ab > 1.