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$ C^1$ spline wavelets on triangulations

Authors: Rong-Qing Jia and Song-Tao Liu
Journal: Math. Comp. 77 (2008), 287-312
MSC (2000): Primary 41A15, 41A63, 42C40, 65D07, 65N30
Published electronically: September 12, 2007
MathSciNet review: 2353954
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Abstract: In this paper we investigate spline wavelets on general triangulations. In particular, we are interested in $ C^1$ wavelets generated from piecewise quadratic polynomials. By using the Powell-Sabin elements, we set up a nested family of spaces of $ C^1$ quadratic splines, which are suitable for multiresolution analysis of Besov spaces. Consequently, we construct $ C^1$ wavelet bases on general triangulations and give explicit expressions for the wavelets on the three-direction mesh. A general theory is developed so as to verify the global stability of these wavelets in Besov spaces. The wavelet bases constructed in this paper will be useful for numerical solutions of partial differential equations.

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Additional Information

Rong-Qing Jia
Affiliation: Department of Mathematical and Statistical Sciences, University of Alberta, Edmonton, Canada T6G 2G1

Song-Tao Liu
Affiliation: Department of Mathematics, University of Chicago, Chicago, Illinois 60637

Keywords: Splines, wavelets, $C^1$ spline wavelets, general triangulations, three-direction meshes, Riesz bases, Sobolev spaces, Besov spaces.
Received by editor(s): July 7, 2004
Received by editor(s) in revised form: November 29, 2006
Published electronically: September 12, 2007
Article copyright: © Copyright 2007 American Mathematical Society
The copyright for this article reverts to public domain 28 years after publication.

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