Remote Access Mathematics of Computation
Green Open Access

Mathematics of Computation

ISSN 1088-6842(online) ISSN 0025-5718(print)



Wavelets on manifolds: An optimized construction

Authors: Angela Kunoth and Jan Sahner
Journal: Math. Comp. 75 (2006), 1319-1349
MSC (2000): Primary 65T60, 54C20; Secondary 42C40, 34B05
Published electronically: May 3, 2006
MathSciNet review: 2219031
Full-text PDF Free Access

Abstract | References | Similar Articles | Additional Information

Abstract: A key ingredient of the construction of biorthogonal wavelet bases for Sobolev spaces on manifolds, which is based on topological isomorphisms is the Hestenes extension operator. Here we firstly investigate whether this particular extension operator can be replaced by another extension operator. Our main theoretical result states that an important class of extension operators based on interpolating boundary values cannot be used in the construction setting required by Dahmen and Schneider. In the second part of this paper, we investigate and optimize the Hestenes extension operator. The results of the optimization process allow us to implement the construction of biorthogonal wavelets from Dahmen and Schneider. As an example, we illustrate a wavelet basis on the 2-sphere.

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

  • [A] R. Adams, Sobolev Spaces, Academic Press, 1975. MR 0450957 (56 #9247)
  • [B] D. Bertsekas, Nonlinear Programming, Athena Scientific, 1999.
  • [Bo] C. de Boor, A Practical Guide to Splines, Applied Mathematical Sciences 27, Springer, 2003. MR 0507062 (80a:65027)
  • [CF] Z. Ciesielski, T. Figiel, Spline bases in classical function spaces on compact $ C^{\infty }$ manifolds: Part I and II, Studia Mathematica, 1983, 1-58 and 95-136. MR 0728195 (85f:46060a) and MR 0730015 (85f:46060b)
  • [CTU] C. Canuto, A. Tabacco, K. Urban, The wavelet element method, part I: Construction and analysis, Appl. Comput. Harm. Anal. 6, 1999, 1-52. MR 1664902 (99k:42055)
  • [DKU] W. Dahmen, A. Kunoth, K. Urban, Biorthogonal spline wavelets on the interval--Stability and moment conditions, Appl. Comput. Harmon. Anal. 6, No.2, 1999, 132-196. MR 1676771 (99m:42046)
  • [DS] W. Dahmen, R. Schneider, Wavelets on manifolds I: Construction and domain decomposition, SIAM J. Math. Anal. 31, No. 1, 1999, 184-230. MR 1742299 (2000k:65242)
  • [DS1] W. Dahmen, R. Schneider, Composite wavelet bases for operator equations, Math. Comp. 68, 1999, 1533-1567. MR 1648379 (99m:65122)
  • [DS2] W. Dahmen, R. Schneider, Wavelets with complementary boundary conditions--function spaces on the cube, Results in Mathematics 34, 1998, 255-293. MR 1652724 (99h:42057)
  • [DSt] W. Dahmen, R. Stevenson, Element-by-element construction of wavelets satisfying stability and moment conditions, SIAM J. Numer. Anal. 37, No. 1, 1999, 319-352. MR 1742747 (2001c:65144)
  • [H] M. R. Hestenes, Extension of the range of a differentiable function, Duke Math. J. 8, 1941, 183-192. MR 0003434 (2,219c)
  • [Ha] H. Harbrecht, Wavelet Galerkin Schemes for the Boundary Element Method in Three Dimensions, Dissertation, Technische Universität Chemnitz, 2001.
  • [IGPM] A. Barinka, T. Bartsch, K. Urban, J. Vorloeper, The Multilevel Library, Institut für Geometrie und Praktische Mathematik, RWTH Aachen, 2001.
  • [KS] A. Kunoth, J. Sahner, Wavelets on manifolds: An optimized construction (extended version), SFB 611 Preprint #163, Universität Bonn, July 2004, revised, April 2005, available at$ \sim$kunoth/papers/papers.html
  • [S] J. Sahner, On the Optimized Construction of Wavelets on Manifolds, Diploma Thesis (in English), Universität Bonn, September 2003.

Similar Articles

Retrieve articles in Mathematics of Computation with MSC (2000): 65T60, 54C20, 42C40, 34B05

Retrieve articles in all journals with MSC (2000): 65T60, 54C20, 42C40, 34B05

Additional Information

Angela Kunoth
Affiliation: Institut für Numerische Simulation, Universität Bonn, Wegelerstr. 6, 53115 Bonn, Germany

Jan Sahner
Affiliation: Konrad-Zuse-Zentrum für Informationstechnik Berlin (ZIB), Takustr. 7, 14195 Berlin, Germany

Keywords: Wavelets on manifolds, topological isomorphisms, extension operators, optimized Hestenes extension, trace dependent operators, biorthogonal wavelets, 2-sphere.
Received by editor(s): July 30, 2004
Received by editor(s) in revised form: April 16, 2005
Published electronically: May 3, 2006
Additional Notes: This work has been supported by the Deutsche Forschungsgemeinschaft (SFB 611) at the Universität Bonn.
Dedicated: Dedicated to Peter Deuflhard on the occasion of his 60th birthday
Article copyright: © Copyright 2006 American Mathematical Society
The copyright for this article reverts to public domain 28 years after publication.

American Mathematical Society