Convergence rates of cascade algorithms

Author:
Rong-Qing Jia

Journal:
Proc. Amer. Math. Soc. **131** (2003), 1739-1749

MSC (2000):
Primary 39B12, 41A25, 42C40, 65D99

DOI:
https://doi.org/10.1090/S0002-9939-03-06953-3

Published electronically:
January 15, 2003

MathSciNet review:
1955260

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Abstract | References | Similar Articles | Additional Information

Abstract: We consider solutions of a refinement equation of the form

where is a finitely supported sequence called the refinement mask. Associated with the mask is a linear operator defined on by . This paper is concerned with the convergence of the cascade algorithm associated with , i.e., the convergence of the sequence in the -norm.

Our main result gives estimates for the convergence rate of the cascade algorithm. Let be the normalized solution of the above refinement equation with the dilation matrix being isotropic. Suppose lies in the Lipschitz space , where and . Under appropriate conditions on , the following estimate will be established:

where and is a constant. In particular, we confirm a conjecture of A. Ron on convergence of cascade algorithms.

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Additional Information

**Rong-Qing Jia**

Affiliation:
Department of Mathematics, University of Alberta, Edmonton, Alberta, Canada T6G 2G1

Email:
jia@xihu.math.ualberta.ca

DOI:
https://doi.org/10.1090/S0002-9939-03-06953-3

Keywords:
Refinement equations,
refinable functions,
cascade algorithms,
subdivision schemes,
rates of convergence

Received by editor(s):
August 29, 2001

Published electronically:
January 15, 2003

Additional Notes:
The author was supported in part by NSERC Canada under Grant OGP 121336

Communicated by:
David R. Larson

Article copyright:
© Copyright 2003
American Mathematical Society