Abstract
In this paper we give an explicit construction of compactly supported prewavelets on differentiable, twodimensional, piecewise polynomial quadratic finite element spaces of L 2 (ℝ2), sampled on the hexagonal grid. The obtained prewavelet basis is stable in the Sobolev spaces \( \mathcal{H}^s \) for \( |s| < \tfrac{5} {2} \). In particular, the prewavelet basis is generated by one single function vector ψ consisting of three generating functions ψ 1,ψ 2,ψ 3 that are globally invariant by a rotation of 2π/3.
This work is partially supported by the Flemish Fund for Scientific Research (FWO Vlaanderen) project MISS (G.0211.02), and by the Belgian Programme on Interuniversity Attraction Poles, initiated by the Belgian Federal Science Policy Office. The scientific responsibility rests with the authors.
This is a preview of subscription content, log in via an institution to check access.
Access this chapter
Tax calculation will be finalised at checkout
Purchases are for personal use only
Preview
Unable to display preview. Download preview PDF.
References
M. D. Buhmann, O. Davydov, and T. N. T. Goodman. Cubic spline prewavelets on the four-directional mesh. Foundations of Computational Mathematics, 3:113–133, 2003.
W. Dahmen and A. Kunoth. Multilevel preconditioning. Numer. Math., 63:315–344, 1992.
W. Dahmen and R. Schneider. Wavelets on manifolds I: Construction and domain decomposition. SIAM J. Math. Anal., 31:184–230, 1999.
W. Dahmen and R. Stevenson. Element-by-element construction of wavelets satisfying stability and moment conditions. SIAM J. Numer. Anal., 37(1):319–352, 2000.
P. Dierckx. On calculating normalized Powell-Sabin B-splines. Comput. Aided Geom. Design, 15(3):61–78, 1997.
G.C. Donovan, J. S. Geronimo, and D. P. Hardin. Intertwining multiresolution analysis and the construction of piecewise polynomial wavelets. SIAM J. Math. Anal., 27:1791–1815, 1996.
G.C. Donovan, J. S. Geronimo, and D. P. Hardin. Compactly supported, piecewise affine scaling functions over triangulations. Constr. Approx., 16:201–219, 2000.
M. S. Floater and E. G. Quak. Piecewise linear prewavelets on arbitrary triangulations. Numer. Math., 82:221–252, 1999.
M. S. Floater and E. G. Quak. Piecewise linear prewavelets over Type-2 triangulations. Comput. Supplement, 14:89–103, 2001.
D. P. Hardin and D. Hong. Construction of wavelets and prewavelets over triangulations. J. Comput. Appl. Math., 155:91–109, 2003.
U. Kotyczka and P. Oswald. Piecewise linear prewavelets of small support. In C. K. Chui and L. L. Schumaker, editors, Approximation Theory VIII, Vol 2, pages 235–242. World Scientific, Singapore, 1995.
J. Maes and A. Bultheel. Stable lifting construction of non-uniform biorthogonal spline wavelets with compact support. TW Report 437, Department of Computer Science, Katholieke Universiteit Leuven, Belgium, October 2005.
J. Maes and A. Bultheel. Stable multiresolution analysis on triangles for surface compression. 2005. Submitted.
J. Maes, E. Vanraes, P. Dierckx, and A. Bultheel. On the stability of normalized Powell-Sabin B-splines. J. Comput. Appl. Math., 170(1):181–196, 2004.
P. Oswald. Multilevel finite element approximation: Theory and applications. B.G. Teubner, Stuttgart, 1994.
M. J. D. Powell and M. A. Sabin. Piecewise quadratic approximations on triangles. ACM Trans. on Math. Software, 3:316–325, 1977.
R. Stevenson. Piecewise linear (pre-)wavelets on non-uniform meshes. In W. Hackbusch and G. Wittum, editors, Multigrid Methods V, pages 306–319. Springer-Verlag, Heidelberg, 1996.
R. Stevenson. Locally supported, piecewise polynomial biorthogonal wavelets on nonuniform meshes. Constr. Approx., 19(4):477–508, 2003.
E. Vanraes, J. Maes, and A. Bultheel. Powell-Sabin spline wavelets. Intl. Journal of Wavelets, Multiresolution and Information Processing, 2(1):23–42, 2004.
E. Vanraes, J. Windmolders, A. Bultheel, and P. Dierckx. Automatic construction of control triangles for subdivided Powell-Sabin splines. Comput. Aided Geom. Design, 21(7):671–682, 2004.
J. Windmolders, E. Vanraes, P. Dierckx, and A. Bultheel. Uniform Powell-Sabin spline wavelets. J. Comput. Appl. Math., 154(1):125–142, 2003.
Author information
Authors and Affiliations
Editor information
Editors and Affiliations
Rights and permissions
Copyright information
© 2006 Birkhäuser Verlag Basel/Switzerland
About this paper
Cite this paper
Maes, J., Bultheel, A. (2006). Powell-Sabin Spline Prewavelets on the Hexagonal Lattice. In: Qian, T., Vai, M.I., Xu, Y. (eds) Wavelet Analysis and Applications. Applied and Numerical Harmonic Analysis. Birkhäuser Basel. https://doi.org/10.1007/978-3-7643-7778-6_21
Download citation
DOI: https://doi.org/10.1007/978-3-7643-7778-6_21
Publisher Name: Birkhäuser Basel
Print ISBN: 978-3-7643-7777-9
Online ISBN: 978-3-7643-7778-6
eBook Packages: Mathematics and StatisticsMathematics and Statistics (R0)