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Proceedings of the American Mathematical Society

Published by the American Mathematical Society, the Proceedings of the American Mathematical Society (PROC) is devoted to research articles of the highest quality in all areas of pure and applied mathematics.

ISSN 1088-6826 (online) ISSN 0002-9939 (print)

The 2020 MCQ for Proceedings of the American Mathematical Society is 0.85.

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Admissible vectors for the regular representation
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by Hartmut Führ PDF
Proc. Amer. Math. Soc. 130 (2002), 2959-2970 Request permission

Abstract:

It is well known that for irreducible, square-integrable representations of a locally compact group, there exist so-called admissible vectors which allow the construction of generalized continuous wavelet transforms. In this paper we discuss when the irreducibility requirement can be dropped, using a connection between generalized wavelet transforms and Plancherel theory. For unimodular groups with type I regular representation, the existence of admissible vectors is equivalent to a finite measure condition. The main result of this paper states that this restriction disappears in the nonunimodular case: Given a nondiscrete, second countable group $G$ with type I regular representation $\lambda _G$, we show that $\lambda _G$ itself (and hence every subrepresentation thereof) has an admissible vector in the sense of wavelet theory iff $G$ is nonunimodular.
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Additional Information
  • Hartmut Führ
  • Affiliation: Zentrum Mathematik, TU München, D-80290 München, Germany
  • Address at time of publication: Institut für Biomathematik und Biometrie, GSF-Forschungszentrum für Umwelt und Gesundheit, Ingolstaedter Landstrasse 1, D-85764 Neuherberg, Germany
  • Email: fuehr@gsf.de
  • Received by editor(s): October 26, 2000
  • Received by editor(s) in revised form: May 3, 2001
  • Published electronically: March 12, 2002
  • Communicated by: David R. Larson
  • © Copyright 2002 American Mathematical Society
  • Journal: Proc. Amer. Math. Soc. 130 (2002), 2959-2970
  • MSC (2000): Primary 43A30; Secondary 42C40
  • DOI: https://doi.org/10.1090/S0002-9939-02-06433-X
  • MathSciNet review: 1908919