Gabor fields and wavelet sets for the Heisenberg group

Abstract

We study singly-generated wavelet systems on ${\mathbb{R}}^2$ that are naturally associated with rank-one wavelet systems on the Heisenberg group $N$. We prove a necessary condition on the generator in order that any such system be a Parseval frame. Given a suitable subset $I$ of the dual of $N$, we give an explicit construction for Parseval frame wavelets that are associated with $I$. We say that $g\in L^2(I\times \mathbb{R})$ is Gabor field over $I$ if, for a.e. $\lambda \in I$, $|\lambda {|}^{1/2}g(\lambda ,\cdot )$ is the Gabor generator of a Parseval frame for $L^2(\mathbb{R})$, and that $I$ is a Heisenberg wavelet set if every Gabor field over $I$ is a Parseval frame (mother-)wavelet for $L^2({\mathbb{R}}^2)$. We then show that $I$ is a Heisenberg wavelet set if and only if $I$ is both translation congruent with a subset of the unit interval and dilation congruent with the Shannon set.