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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
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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.

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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.