h . FA ] 2 4 N ov 1 99 8 Weyl - Heisenberg frames for subspaces of L 2 ( R )

AWeyl-Heisenberg frame {EmbTnag}m,n∈Z = {eg(·−na)}m,n∈Z for L2(R) allows every function f ∈ L2(R) to be written as an infinite linear combination of translated and modulated versions of the fixed function g ∈ L2(R). In the present paper we find sufficient conditions for {EmbTnag}m,n∈Z to be a frame for span{EmbTnag}m,n∈Z , which, in general, might just be a subspace of L2(R) . Even our condition for {EmbTnag}m,n∈Z to be a frame for L2(R) is significantly weaker than the previous known conditions. The results also shed new light on the classical results concerning frames for L2(R), showing for instance that the condition G(x) := ∑ n∈Z |g(x−na)|2 > A > 0 is not necessary for {EmbTnag}m,n∈Z to be a frame for span{EmbTnag}m,n∈Z . Our work is inspired by a recent paper by Benedetto and Li [1], where the relationship between the zero-set of the function G and frame properties of the set of functions {g(· − n)}n∈Z is analyzed. 1 Preliminaries and notation. Let H denote a separable Hilbert space with the inner product < ·, · > linear in the first entry. Let I denote a countable index set. ∗The first author was supported by NSF grant DMS 970618 and the second author by the Danish Research Council. The second author also wants to thank University of Charlotte, NC, and University of Missouri-Columbia, MO, for providing good working conditions. AMS Mathematics subject classification: 42C15 1 Weyl-Heisenberg frame sequences 2 We say that {gi}i∈I ⊆ H is a frame (for H) if there exist constants A,B > 0 such that A||f || ≤ ∑ i∈I | < f, gi > | ≤ B||f ||, ∀f ∈ H. In particular a frame for H is complete, i.e., span{gi}i∈I = H. In case {gi}i∈I is not complete, {gi}i∈I can still be a frame for the subspace span{gi}i∈I ; in that case we say that {gi}i∈I is a frame sequence. The numbers A,B that appear in the definition of a frame are called frame bounds. Orthonormal bases and, more generally, Riesz bases, are frames. Recall that {gi}i∈I is a Riesz basis for H if span{gi}i∈I = H and ∃A,B > 0 : A ∑