Multiple refinable Hermite interpolants
type: Item
instance of: scholarly article
title: Multiple refinable Hermite interpolants
zbMATH Open document ID: 0952.41003
author: Ding-Xuan Zhou
published in: Journal of Approximation Theory
publication date: 2001-01-15T00:00:00Z
review text: We consider solutions of a system of refinement equations written in the form as \(\varphi(x)= \sum_{n\in \mathbb{Z}}a(n) \varphi(2x-n)\), where \(\varphi= (\varphi_1, \dots, \varphi_r)^T\) is a vector of compactly supported functions on \(\mathbb{R}\) and \(a\) is a finitely supported sequence of \(r\times r\) matrices called the refinement mask. If \(\varphi\) is a continuous solution and \(a\) is supported on \([N_1,N_2]\), then \(v:=(\varphi (n))^{N_2-1}_{n=N_1}\) is an eigenvector of the matrix \((a(2k-n))^{N_2-1}_{k,n=N_1}\) associated with eigenvalue 1. Conversely, given such an eigenvector \(v\), we may ask whether there exists a continuous solution \(\varphi\) such that \(\varphi(n)= v(n)\) for \(N_1\leq n\leq N_2-1\) \((\varphi(n)=0\) for \(n\notin [N_1,N_2-1]\), according to the support). The first part of this paper answers this question completely. This existence problem is more general than either the convergence of the subdivision scheme or the requirement of stability, since in one of the latter cases, the eigenvector \(v\) is unique to a constant multiplication. The second part of this paper is concerned with Hermite interpolant solutions, i.e., for some \(n_0\in \mathbb{Z}\) and \(j,m=1, \dots,r\), \(\varphi_j\in C^{r-1} (\mathbb{R})\) and \(\varphi_j^{(m-1)}(n)= \delta_{j,m} \delta_{n,n_0}\), \(n\in\mathbb{Z}\). We provide a necessary and sufficient condition for the refinement equation to have an Hermite interpolant solution. The condition is strictly in terms of the refinement mask. Our method is to characterize the existence and the Hermite interpolant condition by joint spectral radii of matrices. Several concrete examples are presented to illustrate the general theory.
reviewed by: Boris Rubin
Mathematics Subject Classification ID: 41A05
zbMATH DE Number: 1417551
zbMATH Keywords: refinement equations
zbMATH Keywords: Hermite interpolants
zbMATH Keywords: spectral radii
MaRDI profile type: Publication
full work available at URL: jath.1999.3376
OpenAlex ID: W2044229224
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DOI: 10.1006/JATH.1999.3376
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description: scientific article; zbMATH DE number 1417551