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Proceedings of the American Mathematical Society
Proceedings of the American Mathematical Society
ISSN 1088-6826(online) ISSN 0002-9939(print)

 

The wavelet dimension function is the trace function of a shift-invariant system


Authors: Amos Ron and Zuowei Shen
Journal: Proc. Amer. Math. Soc. 131 (2003), 1385-1398
MSC (2000): Primary 42C15; Secondary 42C30
Published electronically: December 6, 2002
MathSciNet review: 1949868
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Abstract: In this note, we observe that the dimension function associated with a wavelet system is the trace of the Gramian fibers of the shift-invariant system generated by the negative dilations of the mother wavelets. When this shift-invariant system is a tight frame, each of the Gramian fibers is an orthogonal projector, and its trace, then, coincides with its rank. This connection leads to simple proofs of several results concerning the dimension function, and the arguments extend to the bi-frame case.


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Additional Information

Amos Ron
Affiliation: Computer Sciences Department, University of Wisconsin-Madison, 1210 West Dayton, Madison, Wisconsin 53706
Email: amos@cs.wisc.edu

Zuowei Shen
Affiliation: Department of Mathematics, National University of Singapore, 10 Kent Ridge Crescent, Singapore 119260
Email: matzuows@leonis.nus.edu.sg

DOI: http://dx.doi.org/10.1090/S0002-9939-02-06677-7
PII: S 0002-9939(02)06677-7
Keywords: Dimension function, frames, multiresolution analysis, wavelets
Received by editor(s): June 8, 2001
Published electronically: December 6, 2002
Additional Notes: This work was supported by the US National Science Foundation under Grants DMS-9872890, DBI-9983114 and ANI-0085984, the U.S. Army Research Office under Contract DAAG55-98-1-0443, and the Strategic Wavelet Program Grant from the National University of Singapore
Communicated by: David R. Larson
Article copyright: © Copyright 2002 American Mathematical Society