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Tight compactly supported wavelet frames
of arbitrarily high smoothness

Authors: Karlheinz Gröchenig and Amos Ron
Journal: Proc. Amer. Math. Soc. 126 (1998), 1101-1107
MSC (1991): Primary 42C15; Secondary 42C30
MathSciNet review: 1443828
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Abstract: Based on Ron and Shen's new method for constructing tight wave-let frames, we show that one can construct, for any dilation matrix, and in any spatial dimension, tight wavelet frames generated by compactly supported functions with arbitrarily high smoothness.

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  • [BDR] C. de Boor, R. DeVore and A. Ron, The structure of shift invariant spaces and applications to approximation theory, J. Functional Anal. 119 (1994), 37-78. Ftp site: file name MR 95g:46050
  • [CD] A. Cohen and I. Daubechies, Non-separable bidimensional wavelets bases, Rev. Math. Iberoamericana, Vol 9 (1993), 51-137. MR 94k:42047
  • [CSS] C.K. Chui, X.L. Shi and J. Stöckler, Affine frames, quasi-affine frames and their duals, CAT Report 372, Texas A&M University, College Station, TX, 77843, June 1996.
  • [DDL] S. Dahlke, W. Dahmen, V. Latour, Smooth refinable functions and wavelets obtained from convolution products. Appl. Comp. Harm. Anal. 2 (1995), 68 - 84. MR 95m:42043
  • [D1] I. Daubechies, Orthonormal bases of compactly supported wavelets, Comm. Pure and Applied Math., 41 (1988), 909-996. MR 90m:42039
  • [D2] I. Daubechies, Ten lectures on wavelets, CBMS-NSF conference series in applied mathematics, Vol 61, SIAM, Philadelphia, 1992. MR 93e:42045
  • [GH] K. Gröchenig and A. Haas, Self-similar lattice tilings. J. Fourier Anal. Appl. 1 (1994), 131-170. MR 96j:52037
  • [GM] K. Gröchenig and W. Madych, Multiresolution analysis, Haar bases, and self-similar tilings. IEEE Trans. Inform. Th. 38(2), (1992), 556-568. MR 93i:42001
  • [RS1] A. Ron and Z. Shen, Affine systems in $L_{2}(\mathbb{R}^d)$, the analysis of the analysis operator, J. Functional Anal., to appear. Ftp site: file
  • [RS2] A. Ron and Z. Shen, Compactly supported tight affine spline frames in $L_{2}(\mathbb{R}^d)$, Math. Comp., to appear. Ftp site: file
  • [S] R. S. Strichartz, Wavelets and self-affine tilings. Constr. Approx. 9 (1993), 327 - 346. MR 94f:42039

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Additional Information

Karlheinz Gröchenig
Affiliation: Department of Mathematics U-9, University of Connecticut, Storrs, Connecticut 06269-3009

Amos Ron
Affiliation: Department of Computer Science, University of Wisconsin-Madison, Madison, Wisconsin 53706

Keywords: Affine systems, frames, tight frames, multiresolution analysis, wavelets
Received by editor(s): September 23, 1996
Additional Notes: This work was supported by the National Science Foundation under Grants DMS-9224748 and DMS-9626319, and by the U.S. Army Research Office under Contract DAAH04-95-1-0089.
Communicated by: Palle E. T. Jorgensen
Article copyright: © Copyright 1998 American Mathematical Society

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