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Mathematics of Computation
Mathematics of Computation
ISSN 1088-6842(online) ISSN 0025-5718(print)


Composite wavelet bases for operator equations

Authors: Wolfgang Dahmen and Reinhold Schneider
Journal: Math. Comp. 68 (1999), 1533-1567
MSC (1991): Primary 65Y20, 68Q25, 65F35, 45L10, 65M99, 76D07
Published electronically: March 10, 1999
MathSciNet review: 1648379
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Abstract: This paper is concerned with the construction of biorthogonal wavelet bases defined on a union of parametric images of the unit $n$-cube. These bases are to satisfy certain requirements imposed by applications to a class of operator equations acting on such domains. This covers also elliptic boundary value problems, although this study is primarily motivated by our previous analysis of wavelet methods for pseudo-differential equations with special emphasis on boundary integral equations. In this case it is natural to model the boundary surface as a union of parametric images of the unit cube. It will be shown how to construct wavelet bases on the surface which are composed of wavelet bases defined on each surface patch. Here the relevant properties are the validity of norm equivalences in certain ranges of Sobolev scales, as well as appropriate moment conditions.

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Additional Information

Wolfgang Dahmen
Affiliation: Institut für Geometrie und Praktische Mathematik, RWTH Aachen, Templergraben 55, 52056 Aachen, Germany

Reinhold Schneider
Affiliation: Fakultät für Mathematik, Technische Universität Chemnitz-Zwickau, 09107 Chemnitz, Germany

PII: S 0025-5718(99)01092-3
Keywords: Biorthogonal wavelets, norm equivalences, boundary element methods, composite multiresolution, multiscale methods for partial differential equations
Received by editor(s): December 20, 1996
Received by editor(s) in revised form: December 12, 1997
Published electronically: March 10, 1999
Additional Notes: The work of the first author has been supported in part by DFG grant Da 117/8-2.
The work of the second author has been supported in part by DFG grant SFB 393.
Article copyright: © Copyright 1999 American Mathematical Society

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