On a conjecture about MRA Riesz wavelet bases

Author:
Bin Han

Journal:
Proc. Amer. Math. Soc. **134** (2006), 1973-1983

MSC (2000):
Primary 42C20, 41A15, 41A05

Published electronically:
December 19, 2005

MathSciNet review:
2215766

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Abstract: Let be a compactly supported refinable function in such that the shifts of are stable and for a -periodic trigonometric polynomial . A wavelet function can be derived from by . If is an orthogonal refinable function, then it is well known that generates an orthonormal wavelet basis in . Recently, it has been shown in the literature that if is a -spline or pseudo-spline refinable function, then always generates a Riesz wavelet basis in . It was an open problem whether can always generate a Riesz wavelet basis in for any compactly supported refinable function in with stable shifts. In this paper, we settle this problem by proving that for a family of arbitrarily smooth refinable functions with stable shifts, the derived wavelet function does not generate a Riesz wavelet basis in . Our proof is based on some necessary and sufficient conditions on the -periodic functions and in such that the wavelet function , defined by , generates a Riesz wavelet basis in .

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Additional Information

**Bin Han**

Affiliation:
Department of Mathematical and Statistical Sciences, University of Alberta, Edmonton, Alberta, Canada T6G 2G1

Email:
bhan@math.ualberta.ca

DOI:
https://doi.org/10.1090/S0002-9939-05-08211-0

Keywords:
Riesz wavelet bases,
refinable functions,
stability

Received by editor(s):
October 1, 2004

Received by editor(s) in revised form:
February 4, 2005

Published electronically:
December 19, 2005

Additional Notes:
This research was supported in part by the Natural Sciences and Engineering Research Council of Canada (NSERC Canada) under Grant G121210654.

Communicated by:
David R. Larson

Article copyright:
© Copyright 2005
American Mathematical Society

The copyright for this article reverts to public domain 28 years after publication.