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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
DOI: https://doi.org/10.1090/S0025-5718-98-00898-9
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
MR Author ID: 292105
Email: matzuows@leonis.nus.sg

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