## Compactly supported tight affine spline frames in $L_2(\mathbb R^d)$

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- by Amos Ron and Zuowei Shen PDF
- Math. Comp.
**67**(1998), 191-207 Request permission

## 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 $\begin {pmatrix}1&1\ 1&-1\end {pmatrix}$.## References

- C. de Boor, K. Höllig, and S. Riemenschneider,
*Box splines*, Applied Mathematical Sciences, vol. 98, Springer-Verlag, New York, 1993. MR**1243635**, DOI 10.1007/978-1-4757-2244-4 - Albert Cohen and Ingrid Daubechies,
*Nonseparable bidimensional wavelet bases*, Rev. Mat. Iberoamericana**9**(1993), no. 1, 51–137. MR**1216125**, DOI 10.4171/RMI/133 - C.K. Chui and X. Shi, Inequalities on matrix-dilated Littlewood-Paley energy functions and oversampled affine operators, SIAM J. Math. Anal.
**28**(1997), 213–232. - Charles K. Chui, Kurt Jetter, and Joachim Stöckler,
*Wavelets and frames on the four-directional mesh*, Wavelets: theory, algorithms, and applications (Taormina, 1993) Wavelet Anal. Appl., vol. 5, Academic Press, San Diego, CA, 1994, pp. 213–230. MR**1321431**, DOI 10.1016/B978-0-08-052084-1.50016-8 - Ingrid Daubechies,
*The wavelet transform, time-frequency localization and signal analysis*, IEEE Trans. Inform. Theory**36**(1990), no. 5, 961–1005. MR**1066587**, DOI 10.1109/18.57199 - Ingrid Daubechies,
*Ten lectures on wavelets*, CBMS-NSF Regional Conference Series in Applied Mathematics, vol. 61, Society for Industrial and Applied Mathematics (SIAM), Philadelphia, PA, 1992. MR**1162107**, DOI 10.1137/1.9781611970104 - Ingrid Daubechies,
*Orthonormal bases of compactly supported wavelets*, Comm. Pure Appl. Math.**41**(1988), no. 7, 909–996. MR**951745**, DOI 10.1002/cpa.3160410705 - Christopher E. Heil and David F. Walnut,
*Continuous and discrete wavelet transforms*, SIAM Rev.**31**(1989), no. 4, 628–666. MR**1025485**, DOI 10.1137/1031129 - Amos Ron and Zuowei Shen,
*Frames and stable bases for shift-invariant subspaces of $L_2(\mathbf R^d)$*, Canad. J. Math.**47**(1995), no. 5, 1051–1094. MR**1350650**, DOI 10.4153/CJM-1995-056-1 - A. Ron and Z. Shen, Affine systems in $L_{2}(\mathbb {R}^d )$: the analysis of the analysis operator, J. Func. Anal., to appear. Ftp site: anonymous@ftp.cs.wisc.edu

## 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
- 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.
- © Copyright 1998 American Mathematical Society
- 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