Construction of compactly supported refinable componentwise polynomial functions in \(\mathbb R^2\)
type: Item
instance of: scholarly article
title: Construction of compactly supported refinable componentwise polynomial functions in \(\mathbb R^2\)
zbMATH Open document ID: 1231.42042
DOI: 10.1155/2011/321428
author: Ning Bi
author: Yan-Mei Xue
published in: Abstract and Applied Analysis
publication date: 2012-01-16T00:00:00Z
review text: A compactly supported function \(f:\,{\mathbb R}^2 \to {\mathbb C}\) is called a componentwise polynomial, if there exists an open set \(G\subset {\mathbb R}^2\) such that \({\mathbb R}^2 \setminus G\) is a set of measure zero and the restriction of \(f\) on any connected open component of \(G\) coincides with some polynomial. Let \(f\in L^1({\mathbb R}^2)\) be a compactly supported function satisfying a refinement equation with a diagonal dilation matrix. The authors give sufficient conditions for the mask symbol of \(f\) such that \(f\) is a componentwise polynomial. The componentwise polynomial is iteratively computed. Examples are presented too.
reviewed by: Manfred Tasche
Mathematics Subject Classification ID: 42C40
Mathematics Subject Classification ID: 65T60
zbMATH DE Number: 5997785
zbMATH Keywords: bivariate refinable function
zbMATH Keywords: compact support
zbMATH Keywords: refinement equation
zbMATH Keywords: diagonal dilation matrix
zbMATH Keywords: mask symbol
zbMATH Keywords: componentwise polynomial
Wikidata QID: Q58654127
MaRDI profile type: Publication
full work available at URL: https://doi.org/10.1155/2011/321428
OpenAlex ID: W2043173764
cites work: Two-Scale Difference Equations II. Local Regularity, Infinite Products of Matrices and Fractals
cites work: Stationary subdivision
cites work: Subdivisions with infinitely supported mask
cites work: Self-affine tiles in \(\mathbb{R}^n\)
cites work: Q4719277
cites work: Classification of refinable splines in \(\mathbb R^{d }\)
cites work: Structure of refinable splines
cites work: Refinable functions with compact support
cites work: Classification of refinable splines
cites work: Characterization of compactly supported refinable splines
cites work: Refinement equations and spline functions
cites work: Eigenvalues of scaling operators and a characterization of $B$-splines
cites work: Construction of symmetric or anti-symmetric B-spline wavelets and their dual wavelets
cites work: Pseudo box splines
cites work: A class of compactly supported refinable componentwise constant functions in \(\mathbb R^2\)
cites work: Asymptotic behavior of \(M\)-band scaling functions of Daubechies type
cites work: Examples of refinable componentwise polynomials
cites work: Componentwise polynomial solutions and distribution solutions of refinement equations
description: scientific article; zbMATH DE number 5997785