Let $a$, $b$ be two fixed non-zero constants. A measurable set $E\subset \mathbb{ R}$ is called a Weyl-Heisenberg frame set for $(a, b)$ if the function $g=\chi_{ E}$ generates a Weyl-Heisenberg frame for $L^2(\mathbb{ R})$ 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 $\{ n_{ 1}<n_{ 2}<...<n_{ k}\}$ 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.
Categories: math.FA