Examples of refinable componentwise polynomials
type: Item
instance of: scholarly article
title: Examples of refinable componentwise polynomials
zbMATH Open document ID: 1114.41005
author: Ning Bi
author: Bin Han
author: Zuowei Shen
published in: Applied and Computational Harmonic Analysis
publication date: 2007-05-03T00:00:00Z
review text: A function \(\phi: {\mathbb R} \to {\mathbb C}\) is called a componentwise polynomal if there exists an open set \(G\) such that the Lebesgue measure of \({\mathbb R}\setminus G\) is zero and the restriction of \(\varphi\) to every connected component of \(G\) is a polynomial. A function \(\varphi\) is called interpolatory if it is continuous, \(\varphi (0)=1\), and \(\varphi (k)=0\) for all integer \(k \neq 0\). A function \(\varphi\) is called orthonormal if \(\{\varphi(\cdot-k)\}\) is an orthonormal system in \(L_2({\mathbb R})\). A function \(\varphi\) is called \(M\)-refinable if \({\widehat \varphi}(M\xi)=H(\xi) {\widehat \varphi}(\xi)\), where \({\widehat \varphi}\) is a Fourier transform of \(\varphi\) and \(H\) is a \(2\pi\)-periodic trigonometric polynomial. The paper presents four examples of compactly supported orthonormal refinable componentwise polynomial functions. In these examples \(M\) equals 3, 6, or 9. The authors note that for \(M=2\) a compactly supported refinable componentwise polynomial must be a B-spline.
Mathematics Subject Classification ID: 41A15
Mathematics Subject Classification ID: 42C40
Mathematics Subject Classification ID: 42C15
Mathematics Subject Classification ID: 41A05
zbMATH DE Number: 5148788
zbMATH Keywords: componentwise polynomial
zbMATH Keywords: symmetric refinable function
zbMATH Keywords: orthogonality
zbMATH Keywords: interpolation
reviewed by: Yuly Makovoz
MaRDI profile type: Publication
full work available at URL: j.acha.2006.09.002
OpenAlex ID: W1970881336
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DOI: 10.1016/J.ACHA.2006.09.002
description: scientific article; zbMATH DE number 5148788