Weyl-Heisenberg Frame Wavelets with Basic Supports

Let $a$, $b$ be two fixed non-zero constants. A measurable set $E\subset $\mathbb{R}$$iscalledaWeyl-Heisenbergframesetfor$(a, b)$ifthefunction$g=\chi_{E}$generatesaWeyl-Heisenbergframefor$L^2($\mathbb{R}$)$ under modulates by $b$ and translates by $a$, i.e., $\{e^{imbt}g(t-na{\}}_{m,n\in \mathbb{Z}}$ is a frame for $L^2(\mathbb{R})$. It is an open question on how to characterize all frame sets for a given pair $(a,b)$ in general. In the case that $a=2\pi$ and $b=1$, a result due to Casazza and Kalton shows that the condition that the set $F={\bigcup }_{j=1}^k([0,2\pi )+2n_j\pi )$ (where $\{n_1<n_2<\mathrm{.}\mathrm{.}\mathrm{.}<n_k\}$ are integers) is a Weyl-Heisenberg frame set for $(2\pi ,1)$ is equivalent to the condition that the polynomial $f(z)={\sum }_{j=1}^k{z}^{n_j}$ does not have any unit roots in the complex plane. In this paper, we show that this result can be generalized to a class of more general measurable sets (called basic support sets) and to set theoretical functions and continuous functions defined on such sets.