Weyl-Heisenberg frames for subspaces of L2 (R)

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