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Proceedings of the American Mathematical Society
Proceedings of the American Mathematical Society
ISSN 1088-6826(online) ISSN 0002-9939(print)

 

A direct integral decomposition of the wavelet representation


Authors: Lek-Heng Lim, Judith A. Packer and Keith F. Taylor
Journal: Proc. Amer. Math. Soc. 129 (2001), 3057-3067
MSC (2000): Primary 65T60, 47N40, 22D20, 22D30; Secondary 22D45, 47L30, 47C05
Published electronically: April 16, 2001
MathSciNet review: 1840112
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Abstract:

In this paper we use the concept of wavelet sets, as introduced by X. Dai and D. Larson, to decompose the wavelet representation of the discrete group associated to an arbitrary $n \times n$ integer dilation matrix as a direct integral of irreducible monomial representations. In so doing we generalize a result of F. Martin and A. Valette in which they show that the wavelet representation is weakly equivalent to the regular representation for the Baumslag-Solitar groups.


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Additional Information

Lek-Heng Lim
Affiliation: Department of Mathematics, Malott Hall, Cornell University, Ithaca, New York 14853-4201
Address at time of publication: Department of Pure Mathematics and Mathematical Statistics, University of Cambridge, Wilberforce Road, Cambridge CB3 0WA, United Kingdom
Email: lekheng@math.cornell.edu

Judith A. Packer
Affiliation: Department of Mathematics, National University of Singapore, 10 Kent Ridge Crescent, Singapore 119260
Email: matjpj@leonis.nus.edu.sg

Keith F. Taylor
Affiliation: Department of Mathematics and Statistics, University of Saskatchewan, 106 Wiggins Road, Saskatoon, Saskatchewan, Canada S7N 5E6
Email: taylor@math.usask.ca

DOI: http://dx.doi.org/10.1090/S0002-9939-01-05928-7
PII: S 0002-9939(01)05928-7
Keywords: Wavelet, wavelet set, group representations
Received by editor(s): November 15, 1999
Received by editor(s) in revised form: February 24, 2000
Published electronically: April 16, 2001
Additional Notes: The third author was supported in part by a grant from NSERC Canada.
Communicated by: David R. Larson
Article copyright: © Copyright 2001 American Mathematical Society