A characterization of the higher dimensional groups associated with continuous wavelets

R. S. Laugesen, N. Weaver, G. L. Weiss, E. N. Wilson

Research output: Contribution to journalArticle

Abstract

A subgroup D of GL (n, <) is said to be admissible if the semidirect product of D and ℝ n, considered as a subgroup of the affine group on ℝ n, admits wavelets ψ ε L 2(ℝ n ) satisfying a generalization of the Calderón reproducing, formula. This article provides a nearly complete characterization of the admissible subgroups D. More precisely, if D is admissible, then the stability subgroup D x for the transpose action of D on ℝ n must be compact for a. e. x ε ℝ n; moreover, if Δ is the modular function of D, there must exist an a ε D such that |det a| ≠ Δ(a). Conversely, if the last condition holds and for a. e. x ε ℝ n there exists an ε > 0 for which the ε-stabilizer D x ε is compact, then D is admissible. Numerous examples are given of both admissible and non-admissible groups.

Original languageEnglish (US)
Pages (from-to)89-102
Number of pages14
JournalJournal of Geometric Analysis
Volume12
Issue number1
DOIs
StatePublished - Dec 1 2002

Keywords

  • admissable dialatin groups
  • continuous wavelets
  • discrete wavelets

ASJC Scopus subject areas

  • Geometry and Topology

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