Monatshefte für Mathematik

, Volume 162, Issue 2, pp 119–142

Gabor fields and wavelet sets for the Heisenberg group

Article

DOI: 10.1007/s00605-009-0159-2

Cite this article as:
Currey, B. & Mayeli, A. Monatsh Math (2011) 162: 119. doi:10.1007/s00605-009-0159-2

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|1/2g(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.

Keywords

WaveletHeisenberg groupGabor frameParseval frameMultiplicity free subspace

Mathematics Subject Classification (2000)

Primary 42C3042C15Secondary 22E27

Copyright information

© Springer-Verlag 2009

Authors and Affiliations

  1. 1.Department of Mathematics and Computer ScienceSaint Louis UniversitySt. LouisUSA
  2. 2.Mathematics DepartmentStony Brook UniversityStony BrookUSA