Stationary subdivision for general scaling matrices
type: Item
instance of: scholarly article
title: Stationary subdivision for general scaling matrices
zbMATH Open document ID: 0978.41011
DOI: 10.1007/PL00004397
author: Vera Latour
author: Jürgen Müller
author: Werner Nickel
published in: Mathematische Zeitschrift
publication date: 1998-08-09T00:00:00Z
review text: Let \(M\in\mathbb{Z}^{s\times s}\) be an expanding scaling matrix (modulus of eigenvalues larger-than 1) and \(a=(a_\alpha)_{\alpha\in \mathbb{Z} ^s}\) a mask (sequence with finite support). Then the corresponding stationary subdivision scheme is the mapping \(S_{a,M}\colon 1_\infty(\mathbb{Z} ^s)\to 1_\infty (\mathbb{Z} ^s)\) defined by \((S_{a,M}\lambda)_\alpha=\sum_{\beta \in\mathbb{Z} ^s} a_{\alpha-M \beta}\lambda_\beta\). The subdivision scheme is said to be uniformly convergent if there exists a function \(f_\lambda \in C({\mathbb{R}}^s)\) such that \(\lim_{m\to\infty} \|(S^m_{a,M}\lambda)-f_{\lambda(M^{-m} \bullet)}\|_{ 1_\infty(\mathbb{Z} ^2)}=0\). Such subdivision schemes are important for geometric design and multivariate wavelet analysis. Most of the results for the classical case \(M=2I\) are easily generalized. However, for the convergence, which is studied in this paper, some specific algebraic tools need to be developed in the case of a general \(M\). The main theorem gives a characterization of convergent subdivision schemes with general \(M\) and compactly supported mask.
Mathematics Subject Classification ID: 41A30
Mathematics Subject Classification ID: 41A63
zbMATH DE Number: 1185943
reviewed by: Adhemar Bultheel
MaRDI profile type: Publication
full work available at URL: pl00004397
OpenAlex ID: W2021298774
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description: scientific article; zbMATH DE number 1185943