A construction of interpolating wavelets

on invariant sets

Authors:
Zhongying Chen, Charles A. Micchelli and Yuesheng Xu

Journal:
Math. Comp. **68** (1999), 1569-1587

MSC (1991):
Primary 41A05, 65D15

DOI:
https://doi.org/10.1090/S0025-5718-99-01110-2

Published electronically:
March 18, 1999

MathSciNet review:
1651746

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Abstract | References | Similar Articles | Additional Information

Abstract: We introduce the concept of a refinable set relative to a family of contractive mappings on a metric space, and demonstrate how such sets are useful to recursively construct interpolants which have a multiscale structure. The notion of a refinable set parallels that of a refinable function, which is the basis of wavelet construction. The interpolation points we recursively generate from a refinable set by a set-theoretic multiresolution are analogous to multiresolution for functions used in wavelet construction. We then use this recursive structure for the points to construct multiscale interpolants. Several concrete examples of refinable sets which can be used for generating interpolatory wavelets are included.

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Additional Information

**Zhongying Chen**

Affiliation:
Department of Scientific Computation, Zhongshan University, Guangzhou 510275, P. R. China

Email:
lnsczy@zsulink.zsu.edu.cn

**Charles A. Micchelli**

Affiliation:
IBM T. J. Watson Research Center, P.O. Box 218, Yorktown Heights, New York 10598-0218

Email:
cam@watson.ibm.com

**Yuesheng Xu**

Affiliation:
Department of Mathematics, North Dakota State University, Fargo, North Dakota 58105

Email:
xu@plains.nodak.edu

DOI:
https://doi.org/10.1090/S0025-5718-99-01110-2

Keywords:
Refinable sets,
set wavelets,
interpolating wavelets

Received by editor(s):
August 11, 1997

Received by editor(s) in revised form:
March 3, 1998

Published electronically:
March 18, 1999

Additional Notes:
The work on this paper as a whole was partially supported by the National Science Foundation under grant DMS-9504780.

The first author was partially supported by the National Natural Science Foundation of China, the American Lingnan Foundation and the Advanced Research Foundation of Zhongshan University; the work of this author was performed during his visit to North Dakota State University in the academic year 1995-1996.

The third author was also supported in part by the Alexander von Humboldt Foundation. Some of his work was performed when he was visiting RWTH-Aachen, Germany.

Dedicated:
Dedicated to Shmuel Winograd on the occasion of his sixtieth birthday with friendship and esteem

Article copyright:
© Copyright 1999
American Mathematical Society