
Published online:
01 May 2009
Published in print:
27 November 2008
Online ISBN:
9780191712340
Print ISBN:
9780198526056
Contents
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8.1 Multiresolution on the interval 8.1 Multiresolution on the interval
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8.1.1 Refinement matrices 8.1.1 Refinement matrices
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8.1.2 Boundary scaling functions 8.1.2 Boundary scaling functions
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8.1.3 Biorthogonal multiresolution 8.1.3 Biorthogonal multiresolution
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Index sets Index sets
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Symmetry Symmetry
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Minimal level Minimal level
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Biorthogonalization Biorthogonalization
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Final statement Final statement
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8.1.4 Refinement matrices 8.1.4 Refinement matrices
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8.1.5 Boundary conditions 8.1.5 Boundary conditions
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8.1.6 Symmetry 8.1.6 Symmetry
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8.2 Wavelets on the interval 8.2 Wavelets on the interval
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8.2.1 Stable completion 8.2.1 Stable completion
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8.2.2 Spline-wavelets on the interval 8.2.2 Spline-wavelets on the interval
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Gauss-type elimination of refinement matrices Gauss-type elimination of refinement matrices
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Biorthogonal wavelet bases Biorthogonal wavelet bases
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8.2.3 Further examples 8.2.3 Further examples
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8.2.4 Dirichlet boundary conditions 8.2.4 Dirichlet boundary conditions
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Index sets Index sets
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8.2.5 Quantitative aspects 8.2.5 Quantitative aspects
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8.2.6 Other constructions on the interval 8.2.6 Other constructions on the interval
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Biorthogonal wavelets Biorthogonal wavelets
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Orthonormal wavelets Orthonormal wavelets
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Other types Other types
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8.2.7 Software for wavelets on the interval 8.2.7 Software for wavelets on the interval
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8.2.8 Numerical experiments 8.2.8 Numerical experiments
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Adaptive operator application: APPLY Adaptive operator application: APPLY
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8.2.8.2 Adaptive solution scheme: ELLSOLVE 8.2.8.2 Adaptive solution scheme: ELLSOLVE
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8.3 Tensor product wavelets 8.3 Tensor product wavelets
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8.4 The Wavelet Element Method (WEM) 8.4 The Wavelet Element Method (WEM)
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8.4.1 Matching in 1D 8.4.1 Matching in 1D
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8.4.1.1 Scaling functions 8.4.1.1 Scaling functions
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Wavelets Wavelets
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Additional features Vanishing at the interface Additional features Vanishing at the interface
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8.4.1.3 The influence of the mapping 8.4.1.3 The influence of the mapping
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8.4.2 The setting in arbitrary dimension 8.4.2 The setting in arbitrary dimension
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Domain decomposition and parametric mappings Domain decomposition and parametric mappings
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Multiresolution and wavelets on the subdomains Multiresolution and wavelets on the subdomains
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8.4.2.3 Multiresolution on the global domain 8.4.2.3 Multiresolution on the global domain
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Wavelets on the global domain Wavelets on the global domain
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8.4.2.5 Characterization of Sobolev spaces 8.4.2.5 Characterization of Sobolev spaces
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Vanishing moments Vanishing moments
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8.4.2.6 Characterization in terms of the modified inner product 8.4.2.6 Characterization in terms of the modified inner product
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8.4.3 The WEM in the two-dimensional case 8.4.3 The WEM in the two-dimensional case
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Matching across a side Matching across a side
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8.4.3.2 Interior cross-point 8.4.3.2 Interior cross-point
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Cross-point matching. Cross-point matching.
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Side matching. Side matching.
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Orthogonality to the dual scaling function. Orthogonality to the dual scaling function.
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Full matching conditions. Full matching conditions.
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Biorthogonalization. Biorthogonalization.
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Tensor products of matched univariate functions Tensor products of matched univariate functions
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8.4.3.4 Matching around a boundary cross point 8.4.3.4 Matching around a boundary cross point
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Cross-point matching. Cross-point matching.
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Side matching. Side matching.
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Orthogonality to the dual scaling function. Orthogonality to the dual scaling function.
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Boundary conditions. Boundary conditions.
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8.4.4 Trivariate matched wavelets 8.4.4 Trivariate matched wavelets
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8.4.5 Software for the WEM 8.4.5 Software for the WEM
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8.5 Embedding methods 8.5 Embedding methods
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8.6 Exercises and programs 8.6 Exercises and programs
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Programs Programs
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Chapter
8 Wavelets on General Domains
Get access
Pages
257–393
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Published:November 2008
Cite
Urban, Karsten, 'Wavelets on General Domains', Wavelet Methods for Elliptic Partial Differential Equations, Numerical Mathematics and Scientific Computation (Oxford , 2008; online edn, Oxford Academic, 1 May 2009), https://doi.org/10.1093/acprof:oso/9780198526056.003.0008, accessed 24 Apr. 2025.
Abstract
The construction of wavelets on general domains is performed in three steps. This chapter starts by introducing the construction of scaling functions and wavelets on bounded univariate intervals. Next, building tensor products allows constructing wavelets on rectangular domains. Finally, the Wavelet Element Method (WEM) is introduced. Using non-overlapping domain decomposition and mapping to the unit cube the WEM matches scaling functions and wavelets across the interfaces of the subdomains in order to obtain a globally continuous basis. The realization of the construction in terms of software is shown as well how to use this software.
Keywords:
wavelets on the interval, Wavelet Element Method, tensor product, refinement matrices, norm equivalences
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