Properties of locally linearly independent refinable function vectors
type: Item
instance of: scholarly article
title: Properties of locally linearly independent refinable function vectors
zbMATH Open document ID: 1031.41020
DOI: 10.1016/S0021-9045(03)00036-4
author: Ding-Xuan Zhou
author: Gerlind Plonka
published in: Journal of Approximation Theory
publication date: 2003-06-09T00:00:00Z
review text: Let \(\Phi= (\phi_1,\dots,\phi_r)^T\) be a vector of compactly supported integrable functions on \(\mathbb{R}\) such that \[ \Phi(t)= \sum_{k\in\mathbb{Z}} A(k)\Phi(2t- k)\quad\text{for all }t\in \mathbb{R}, \] where \(\{A(k)\}\) is a finitely supported sequence of \((r\times r)\)-matrices. Under the assumption that \(\Phi\) is locally linearly independent, the following results are established: (1) the global supports of the components of \(\Phi\) start and end with special rational numbers; (2) in contrast with the scalar case, the components of \(\Phi\) may possess a hole in their global support when \(r\geq 2\); (3) if \(A(k)= 0\) for \(k< 0\) and \(k> N\), and if \(A(0)\) and \(A(N)\) do not contain zero rows, then all nonzero components of \(\Phi\) have the global support \([0,N]\) and, in particular, they have no holes. Furthermore, in the case \(r= 2\) there are given two interesting globally linearly independent functions \(\Phi\): (1) a function which is linearly dependent over \((0,1)\) and for which the shift-invariant space generated by \(\Phi\) does not possess a linearly independent basis over \((0,1)\); (2) a function which is linearly dependent over any finite interval.
Mathematics Subject Classification ID: 41A99
Mathematics Subject Classification ID: 41A25
zbMATH DE Number: 1924729
zbMATH Keywords: Refinable function vectors
zbMATH Keywords: Local linear independence
zbMATH Keywords: Global linear
zbMATH Keywords: independence
zbMATH Keywords: Support of refinable functions
reviewed by: Wolfgang W. Breckner
MaRDI profile type: Publication
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description: scientific article; zbMATH DE number 1924729