Local linear independence of the translates of a box spline
type: Item
instance of: scholarly article
title: Local linear independence of the translates of a box spline
zbMATH Open document ID: 0586.41007
DOI: 10.1007/BF01890029
author: Rong-Qing Jia
published in: Constructive Approximation
publication date: 1985-01-01T00:00:00Z
review text: Let \(\Xi =(\xi_ i)^ n_ 1\) be a sequence of vectors in \(R^ m\). The box spline \(M_{\Xi}\) is defined as the distribution given by \[ M_{\Xi}: \phi \to \int_{[0,1]^ n}\phi (\sum^{n}_{i=1}\lambda (i)\xi_ i)d\lambda,\quad \phi \in C_ c^{\infty}(R^ m). \] Suppose that \(\Xi\) contains a basis for \(R^ m\). Then \(M_{\Xi}\in L_{\infty}(R^ m)\). Assume \(\Xi \subset V:=Z^ m\). Consider the translates \(Mv:=M_{\Xi}(\cdot -v)\), \(v\in V\). It is known that \((Mv)_ V\) is linearly dependent unless (*) \(| \det Z| =1\) for all bases \(Z\subset \Xi\). This paper demonstrates that under condition (*), \((Mv)_ V\) is locally linearly independent, i.e., \(\{\) Mv;supp Mv\(\cap A\neq \emptyset \}\) is linearly independent over any open set A.
Mathematics Subject Classification ID: 41A15
Mathematics Subject Classification ID: 41A63
Mathematics Subject Classification ID: 15A04
zbMATH DE Number: 3939905
zbMATH Keywords: box spline
zbMATH Keywords: distribution
MaRDI profile type: Publication
cites work: Q4112369
cites work: Approximation by Smooth Multivariate Splines
cites work: B-splines from parallelepipeds
cites work: Bivariate box splines and smooth pp functions on a three direction mesh
cites work: Translates of multivariate splines
cites work: Linear independence of translates of a box spline
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description: scientific article; zbMATH DE number 3939905