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Transactions of the American Mathematical Society
Transactions of the American Mathematical Society
ISSN 1088-6850(online) ISSN 0002-9947(print)

 

Characterization of Smoothness
of Multivariate Refinable Functions
in Sobolev Spaces


Author: Rong-Qing Jia
Journal: Trans. Amer. Math. Soc. 351 (1999), 4089-4112
MSC (1991): Primary 42C15, 39B99, 46E35
Published electronically: July 1, 1999
MathSciNet review: 1473444
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Abstract | References | Similar Articles | Additional Information

Abstract: Wavelets are generated from refinable functions by using multiresolution analysis. In this paper we investigate the smoothness properties of multivariate refinable functions in Sobolev spaces. We characterize the optimal smoothness of a multivariate refinable function in terms of the spectral radius of the corresponding transition operator restricted to a suitable finite dimensional invariant subspace. Several examples are provided to illustrate the general theory.


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Additional Information

Rong-Qing Jia
Affiliation: Department of Mathematical Sciences, University of Alberta, Edmonton, Canada T6G 2G1
Email: jia@xihu.math.ualberta.ca

DOI: http://dx.doi.org/10.1090/S0002-9947-99-02185-6
PII: S 0002-9947(99)02185-6
Keywords: Refinement equations, refinable functions, wavelets, smoothness, regularity, approximation order, Sobolev spaces, Lipschitz spaces, subdivision operators, transition operators
Received by editor(s): June 11, 1996
Received by editor(s) in revised form: April 14, 1997
Published electronically: July 1, 1999
Additional Notes: Supported in part by NSERC Canada under Grant OGP 121336
Article copyright: © Copyright 1999 American Mathematical Society