Weyl-Heisenberg frames for subspaces of $L^2(R)$

Abstract

A Weyl-Heisenberg frame {E mb Tnag} m,n∈Z = {e 2πimb(·) g(· − na)} m,n∈Z for L 2 (R) allows every function f ∈ L 2 (R) to be written as an infinite linear combination of translated and modulated versions of the fixed function g ∈ L 2 (R). In the present paper we find sufficient conditions for {E mb Tnag} m,n∈Z to be a frame for span{E mb Tnag} m,n∈Z , which, in general, might just be a subspace of L 2 (R). Even our condition for {E mb Tnag} m,n∈Z to be a frame for L 2 (R) is significantly weaker than the previous known conditions. The re-sults also shed new light on the classical results concerning frames for L 2 (R), showing for instance that the condition G(x) := n∈Z |g(x − na)| 2 > A > 0 is not necessary for {E mb Tnag} m,n∈Z to be a frame for span{E mb Tnag} m,n∈Z . Our work is inspired by a recent paper by Benedetto and Li, where the rela-tionship between the zero-set of the function G and frame properties of the set of functions {g(· − n)} n∈Z is analyzed.