Gabor fields and wavelet sets for the Heisenberg group

TL;DR

This paper explores the construction of wavelet systems on the Heisenberg group, establishing conditions for Parseval frames and introducing the concepts of Gabor fields and Heisenberg wavelet sets with explicit characterizations.

Contribution

It provides a necessary condition for generators to form Parseval frames and characterizes Heisenberg wavelet sets through translation and dilation congruences.

Findings

Characterization of Gabor fields over specific sets.

Explicit construction of Parseval frame wavelets.

Equivalence of Heisenberg wavelet sets with certain congruence conditions.

Abstract

We study singly-generated wavelet systems on $\Bbb R^2$ 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 $g\in L^2(I\times \Bbb R)$ 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 $L^2(\Bbb R)$, and that $I$ is a Heisenberg wavelet set if every Gabor field over $I$ is a Parseval frame (mother-)wavelet for $L^2(\Bbb R^2)$. 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.