The construction of wavelets adapted to compact domains
type: Item
instance of: scholarly article
title: The construction of wavelets adapted to compact domains
zbMATH Open document ID: 1271.65161
author: Dale W. Struble
published in: Advances in Computational Mathematics
publication date: 2013-05-06T00:00:00Z
review text: Let \(E\) be a compact set in \(\mathbb R^d\) and \(\rho\) be a measure defined on \(E\). Under some assumptions on the structure of \(E\) and \(\rho\), the author provides a methodology to construct wavelets to the space \(L^2(E, \rho)\). The methodology includes four steps: defining the measure \(\rho\) on \(E\); finding a map from \(E\) onto the unit in \(\mathbb R^d\); constructing the maps along with a refinable vector field; and constructing the initial wavelet space. Examples of the construction for a rectangular surface, a triangular surface, and a smooth simplex in \(\mathbb R^{3}\) are given. An extension of the theory to triangulated manifolds of finite distortion in \(n\) dimensions is also explained.
Mathematics Subject Classification ID: 65T60
Mathematics Subject Classification ID: 14R15
Mathematics Subject Classification ID: 54E45
Mathematics Subject Classification ID: 37C10
Mathematics Subject Classification ID: 53D05
Mathematics Subject Classification ID: 42C40
zbMATH DE Number: 6160928
zbMATH Keywords: wavelets
zbMATH Keywords: fractals
zbMATH Keywords: manifolds
zbMATH Keywords: compact domain
zbMATH Keywords: Jacobian problem
zbMATH Keywords: rectangular surface
zbMATH Keywords: triangular surface
zbMATH Keywords: smooth simplex
zbMATH Keywords: triangulated manifold
reviewed by: Yuri A. Farkov
MaRDI profile type: Publication
full work available at URL: s10444-011-9250-z
OpenAlex ID: W2010575882
cites work: On a partial differential equation involving the Jacobian determinant
cites work: Q3714302
cites work: Change of variables formula under minimal assumptions
cites work: Geometric function theory and nonlinear analysis
cites work: Q4326436
cites work: Continuous refinable functions and self similarity
cites work: Subdivision schemes for iterated function systems
cites work: Using the matrix refinement equation for the construction of wavelets on invariant sets
cites work: Reconstruction and decomposition algorithms for biorthogonal multiwavelets
cites work: Q4485837
cites work: Regularity of the inverse of spatial mappings with finite distortion
DOI: 10.1007/S10444-011-9250-Z
description: scientific article; zbMATH DE number 6160928