Admissible vectors for the regular representation

Author:
Hartmut Führ

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

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 with type I regular representation , we show that itself (and hence every subrepresentation thereof) has an admissible vector in the sense of wavelet theory iff 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

DOI:
https://doi.org/10.1090/S0002-9939-02-06433-X

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