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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
DOI: https://doi.org/10.1090/S0002-9939-05-08211-0
Published electronically: December 19, 2005
MathSciNet review: 2215766
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Abstract: Let $ \phi$ be a compactly supported refinable function in $ L_2(\mathbb{R})$ such that the shifts of $ \phi$ are stable and $ \hat\phi(2\xi)=\hat a(\xi)\hat \phi(\xi)$ for a $ 2\pi$-periodic trigonometric polynomial $ \hat a$. A wavelet function $ \psi$ can be derived from $ \phi$ by $ \hat \psi(2\xi):=e^{-i\xi}\overline{\hat a(\xi+\pi)} \hat \phi(\xi)$. If $ \phi$ is an orthogonal refinable function, then it is well known that $ \psi$ generates an orthonormal wavelet basis in $ L_2(\mathbb{R})$. Recently, it has been shown in the literature that if $ \phi$ is a $ B$-spline or pseudo-spline refinable function, then $ \psi$ always generates a Riesz wavelet basis in $ L_2(\mathbb{R})$. It was an open problem whether $ \psi$ can always generate a Riesz wavelet basis in $ L_2(\mathbb{R})$ for any compactly supported refinable function in $ L_2(\mathbb{R})$ 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 $ \psi$ does not generate a Riesz wavelet basis in $ L_2(\mathbb{R})$. Our proof is based on some necessary and sufficient conditions on the $ 2\pi$-periodic functions $ \hat a$ and $ \hat b$ in $ C^{\infty}(\mathbb{R})$ such that the wavelet function $ \psi$, defined by $ \hat \psi(2\xi):=\hat b(\xi)\hat \phi(\xi)$, generates a Riesz wavelet basis in $ L_2(\mathbb{R})$.


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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.

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