Article

Monatshefte für Mathematik

, Volume 162, Issue 2, pp 119-142

Gabor fields and wavelet sets for the Heisenberg group

  • Bradley CurreyAffiliated withDepartment of Mathematics and Computer Science, Saint Louis University Email author 
  • , Azita MayeliAffiliated withMathematics Department, Stony Brook University

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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.

Keywords

Wavelet Heisenberg group Gabor frame Parseval frame Multiplicity free subspace

Mathematics Subject Classification (2000)

Primary 42C30 42C15 Secondary 22E27