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

A Weyl-Heisenberg frame

for $L^2(R)$ allows every function $f \in L^2(R)$ to be written as an infinite linear combination of translated and modulated versions of the fixed function $g \in L^2(R)$. In the present paper we find sufficient conditions for $\{E_{mb}T_{na}g \}_{m,n \in Z}$ to be a frame for $\overline{span}\{E_{mb}T_{na}g \}_{m,n \in Z}$, which, in general, might just be a subspace of $L^2(R)$. Even our condition for $\{E_{mb}T_{na}g \}_{m,n \in Z}$ 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 $G(x):= \sum_{n \in Z} \vert g(x-na)\vert^2 >A>0$is not necessary for $\{E_{mb}T_{na}g \}_{m,n \in Z}$ to be a frame for $\overline{span} \{E_{mb}T_{na}g \}_{m,n \in Z}$. 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 $\{g( \cdot - n) \}_{n \in Z}$ is analyzed.