Remote Access Mathematics of Computation
Green Open Access

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
Full-text PDF

Abstract | References | Similar Articles | Additional Information

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 [Enhancements On Off] (What's this?)

  • [BHR] C. de Boor, K. Höllig and S.D. Riemenschneider, Box splines, Springer Verlag, New York, (1993). MR 94k:65004
  • [CD] A. Cohen and I. Daubechies, Non-separable bidimensional wavelets bases, Rev. Mat. Iberoamericana, Vol 9 (1993), 51-137. MR 94k:42047
  • [CS] 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. CMP 97:06
  • [CJS] C.K. Chui, K. Jetter, and J. Stöckler, Wavelets and frames on the four-directional mesh, in Wavelets: Theory, Algorithms, and Applications, C. K. Chui, L. Montefusco, L. Puccio (eds.), Academic Press, San Diego, 1994, 213-230. MR 96b:41017
  • [D1] I. Daubechies, The wavelet transform, time-frequency localization and signal analysis, IEEE Trans. Inform. Theory 36 (1990), 961-1005. MR 91e:42038
  • [D2] I. Daubechies, Ten lectures on wavelets, CBMS-NSF conference series in applied mathematics, Vol 61, SIAM, Philadelphia, 1992. MR 93e:42045
  • [D3] I. Daubechies, Orthonormal bases of compactly supported wavelets, Comm. Pure and Appl. Math. 41 (1988), 909-996. MR 90m:42039
  • [HW] C. Heil and D. Walnut, Continuous and discrete wavelet transforms, SIAM Review 31 (1989), 628-666. MR 91c:42032
  • [RS1] A. Ron and Z. Shen, Frames and stable bases for shift-invariant subspaces of $L_{2}(\mathbb R^d)$, Canad. J. Math. 47 (1995), 1051-1094. Ftp site: MR 96k:42049
  • [RS2] 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:

Similar Articles

Retrieve articles in Mathematics of Computation of the American Mathematical Society with MSC (1991): 42C15, 41A15, 41A63, 42C30

Retrieve articles in all journals with MSC (1991): 42C15, 41A15, 41A63, 42C30

Additional Information

Amos Ron
Affiliation: Computer Science Department, University of Wisconsin-Madison, 1210 West Dayton Street, Madison, Wisconsin 53706

Zuowei Shen
Affiliation: Department of Mathematics, National University of Singapore, 10 Kent Ridge Crescent, Singapore 119260

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

American Mathematical Society