, Volume 162, Issue 2, pp 119-142

Gabor fields and wavelet sets for the Heisenberg group

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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.

Communicated by K. Gröchenig.
A. Mayeli was partially supported by the Marie Curie Excellence Team Grant MEXT-CT-2004-013477, Acronym MAMEBIA, funded by the European Commission.