The wavelet dimension function is the trace function of a shift-invariant system
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- by Amos Ron and Zuowei Shen PDF
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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.References
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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
- MR Author ID: 292105
- Email: matzuows@leonis.nus.edu.sg
- 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
- © Copyright 2002 American Mathematical Society
- Journal: Proc. Amer. Math. Soc. 131 (2003), 1385-1398
- MSC (2000): Primary 42C15; Secondary 42C30
- DOI: https://doi.org/10.1090/S0002-9939-02-06677-7
- MathSciNet review: 1949868