Roots of complex polynomials and Weyl-Heisenberg frame sets

Abstract:

A Weyl-Heisenberg frame for $L^{2}(\mathbb R)$ is a frame consisting of modulates $E_{mb}g(t) = e^{2{\pi }imbt}g(t)$ and translates $T_{na}g(t) = g(t-na)$, $m,n\in \mathbb Z$, of a fixed function $g\in L^{2} (\mathbb R)$, for $a,b\in \mathbb R$. A fundamental question is to explicitly represent the families $(g,a,b)$ so that $(E_{mb}T_{na}g)_{m,n\in \mathbb Z}$ is a frame for $L^{2}(\mathbb R)$. We will show an interesting connection between this question and a classical problem of Littlewood in complex function theory. In particular, we show that classifying the characteristic functions ${\chi }_{E}$ for which $(E_{m}T_{n}{\chi }_{E})_{m,n\in \mathbb Z}$ is a frame for $L^{2}(\mathbb R)$ is equivalent to classifying the integer sets $\{n_{1}<n_{2}<\cdots <n_{k}\}$ so that $f(z) = \sum _{j=1}^{k} z^{n_{i}}$ does not have any zeroes on the unit circle in the plane.