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Mathematics of Computation
Mathematics of Computation
ISSN 1088-6842(online) ISSN 0025-5718(print)

 

Compactly supported tight affine
spline frames in $L_{2}(\mathbb{R}^{d})$


Authors: Amos Ron and Zuowei Shen
Journal: Math. Comp. 67 (1998), 191-207
MSC (1991): Primary 42C15, 41A15, 41A63; Secondary 42C30
MathSciNet review: 1433269
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Abstract: The theory of fiberization is applied to yield compactly supported tight affine frames (wavelets) in $L_{2}(\mathbb{R}^{d})$ from box splines. The wavelets obtained are smooth piecewise-polynomials on a simple mesh; furthermore, they exhibit a wealth of symmetries, and have a relatively small support. The number of ``mother wavelets'', however, increases with the increase of the required smoothness.

Two bivariate constructions, of potential practical value, are highlighted. In both, the wavelets are derived from four-direction mesh box splines that are refinable with respect to the dilation matrix $\begin{pmatrix}1&1\\ 1&-1\end{pmatrix} $.


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Additional Information

Amos Ron
Affiliation: Computer Science Department, University of Wisconsin-Madison, 1210 West Dayton Street, 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.sg

DOI: http://dx.doi.org/10.1090/S0025-5718-98-00898-9
PII: S 0025-5718(98)00898-9
Keywords: Affine systems, box splines, four-direction mesh, frames, tight frames, multiresolution analysis, wavelets
Received by editor(s): February 19, 1996
Received by editor(s) in revised form: August 21, 1996
Additional Notes: This work was supported by the National Science Foundation under Grants DMS-9102857, DMS-9224748, and by the U.S. Army Research Office under Contracts DAAL03-G-90-0090, DAAH04-95-1-0089.
Article copyright: © Copyright 1998 American Mathematical Society