Interpolation operators associated with sub-frame sets

Author:
Deguang Han

Journal:
Proc. Amer. Math. Soc. **131** (2003), 275-284

MSC (2000):
Primary 42C15, 47B38

DOI:
https://doi.org/10.1090/S0002-9939-02-06658-3

Published electronically:
June 3, 2002

MathSciNet review:
1929047

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Abstract: Interpolation operators associated with wavelets sets introduced by Dai and Larson play an important role in their operator algebraic approach to wavelet theory. These operators are also related to the von Neumann subalgebras in the ``local commutant'' space, which provides the parametrizations of wavelets. It is a particularly interesting question of how to construct operators which are in the local commutant but not in the commutant. Motivated by some questions about interpolation family and C*-algebras in the local commutant, we investigate the interpolation partial isometry operators induced by sub-frame sets. In particular we introduce the -congruence domain of the associated mapping between two sub-frame sets and use it to characterize these partial isometries in the local commutant. As an application, we obtain that if two wavelet sets have the same -congruence domain, then one is a multiresolution analysis (MRA) wavelet set which implies that the other is also an MRA wavelet set.

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Additional Information

**Deguang Han**

Affiliation:
Department of Mathematics, University of Central Florida, Orlando, Florida 32816

Email:
dhan@pegasus.cc.ucf.edu

DOI:
https://doi.org/10.1090/S0002-9939-02-06658-3

Keywords:
Wavelet,
sub-frame set,
interpolation operators,
congruence domain,
multiresolution analysis,
MRA wavelet set

Received by editor(s):
February 1, 2001

Received by editor(s) in revised form:
September 10, 2001

Published electronically:
June 3, 2002

Communicated by:
David R. Larson

Article copyright:
© Copyright 2002
American Mathematical Society