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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: 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