Some characterizations for box spline wavelets and linear Diophantine equations
type: Item
instance of: scholarly article
title: Some characterizations for box spline wavelets and linear Diophantine equations
zbMATH Open document ID: 0946.42024
published in: Rocky Mountain Journal of Mathematics
publication date: 1999-10-17T00:00:00Z
full work available at URL: CONT28-4.html
review text: For univariate compactly supported distributions satisfying a refinement equation \[ \varphi=\sum_{\alpha\in {\mathbb Z}}b_\alpha \varphi (k \cdot -\alpha) \] with \(2\leq k\in{\mathbb N}\), \textit{R.-Q. Jia} and \textit{J. Wang} [Proc. Am. Math. Soc. 117, No. 4, 1115-1124 (1993; Zbl 0770.42018)] presented necessary and sufficient conditions for linear independence of integer translates of \(\varphi\) in terms of the refinement mask of \(\varphi\). In the multivariate case these conditions remain to be necessary. However, the author gives examples of box splines, where the sufficiency of these conditions does not hold any longer. On the other hand, if the box spline \(M_{\Xi}\) in \({\mathbb R}^s\) satisfies rank \(\Xi=s\), then a direct extension of the criterion for linear independence of refinable distributions holds. This approach hence opens a new way of deriving the linear independence of translates of box splines by means of their refinement masks.
Mathematics Subject Classification ID: 42C40
Mathematics Subject Classification ID: 41A63
Mathematics Subject Classification ID: 41A15
zbMATH DE Number: 1309891
zbMATH Keywords: linear independence
zbMATH Keywords: box spline wavelets
zbMATH Keywords: refinement equation
zbMATH Keywords: linear Diophantine equations
zbMATH Keywords: refinement masks
author: Ding-Xuan Zhou
reviewed by: Gerlind Plonka
MaRDI profile type: Publication
OpenAlex ID: W2025623334
cites work: Q4022923
cites work: Ondelettes, analyses multirésolutions et filtres miroirs en quadrature. (Wavelets, multiscale analysis and quadrature mirror filters)
cites work: Biorthogonal bases of compactly supported wavelets
cites work: On Linear Dependence Relations for Integer Translates of Compactly Supported Distributions
cites work: Translates of multivariate splines
cites work: Algebraic properties of discrete box splines
cites work: Ten Lectures on Wavelets
cites work: Orthonormal bases of compactly supported wavelets
cites work: Linear independence of translates of a box spline
cites work: Multivariate Discrete Splines and Linear Diophantine Equations
cites work: Multiresolution and wavelets
cites work: Stability and Linear Independence Associated with Wavelet Decompositions
cites work: Q3995454
cites work: Wavelets and pre-wavelets in low dimensions
cites work: A necessary and sufficient condition for the linear independence of the integer translates of a compactly supported distribution
cites work: Q4027563
cites work: Stability of Refinable Functions, Multiresolution Analysis, and Haar Bases
cites work: Construction of real-valued wavelets by symmetry
cites work: Linear dependence relations in wavelets and tilings
DOI: 10.1216/RMJM/1181071729
description: scientific article; zbMATH DE number 1309891