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Admissible vectors for the regular representation

Author: Hartmut Führ
Journal: Proc. Amer. Math. Soc. 130 (2002), 2959-2970
MSC (2000): Primary 43A30; Secondary 42C40
Published electronically: March 12, 2002
MathSciNet review: 1908919
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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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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

Keywords: Continuous wavelet transforms, coherent states, square-integrable representations, Plancherel theory, cyclic vectors
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
Article copyright: © Copyright 2002 American Mathematical Society