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. ${\lambda \in I}$, |λ|^1/2 g(λ, ·) 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.