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Wavelets with patchwise cancellation properties


Authors: Helmut Harbrecht and Rob Stevenson
Journal: Math. Comp. 75 (2006), 1871-1889
MSC (2000): Primary Prmary, 46B15, 46E35, 65N55, 65T60
DOI: https://doi.org/10.1090/S0025-5718-06-01867-9
Published electronically: June 22, 2006
MathSciNet review: 2240639
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Abstract: We construct wavelets on general $ n$-dimensional domains or manifolds via a domain decomposition technique, resulting in so-called composite wavelets. With this construction, wavelets with supports that extend to more than one patch are only continuous over the patch interfaces. Normally, this limited smoothness restricts the possibility for matrix compression, and with that the application of these wavelets in (adaptive) methods for solving operator equations. By modifying the scaling functions on the interval, and with that on the $ n$-cube that serves as parameter domain, we obtain composite wavelets that have patchwise cancellation properties of any required order, meaning that the restriction of any wavelet to each patch is again a wavelet. This is also true when the wavelets are required to satisfy zeroth order homogeneous Dirichlet boundary conditions on (part of) the boundary. As a result, compression estimates now depend only on the patchwise smoothness of the wavelets that one may choose. Also taking stability into account, our composite wavelets have all the properties for the application to the (adaptive) solution of well-posed operator equations of orders $ 2 t$ for $ t \in (-\frac{1}{2},\frac{3}{2})$.


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Additional Information

Helmut Harbrecht
Affiliation: Institute of Computer Science and Applied Mathematics, Christian-Albrechts-University of Kiel, Olshausenstr. 40, 24098 Kiel, Germany
Email: hh@numerik.uni-kiel.de

Rob Stevenson
Affiliation: Department of Mathematics, Utrecht University, P.O. Box 80.010, NL-3508 TA Utrecht, The Netherlands
Email: stevenson@math.uu.nl

DOI: https://doi.org/10.1090/S0025-5718-06-01867-9
Keywords: Wavelets, domain decomposition, patchwise cancellation properties, matrix compression, Riesz bases
Received by editor(s): October 8, 2004
Received by editor(s) in revised form: March 14, 2005
Published electronically: June 22, 2006
Additional Notes: This work was supported by the Netherlands Organization for Scientific Research and by the EC-IHP project “Breaking Complexity”
Article copyright: © Copyright 2006 American Mathematical Society
The copyright for this article reverts to public domain 28 years after publication.

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