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.