Characterising pairs of refinable splines
type: Item
instance of: scholarly article
title: Characterising pairs of refinable splines
zbMATH Open document ID: 0894.41012
author: Tim N. T. Goodman
published in: Journal of Approximation Theory
publication date: 1998-08-13T00:00:00Z
review text: In a recent paper by \textit{W. Lawton, S. L. Lee}, and \textit{Z. Shen} [Adv. Comput. Math. 3, No. 1-2, 137-145 (1995; Zbl 0828.41006)] a characterization of refinable spline functions with compact support was presented, and it was shown that the integer translates of such a function can only form a Riesz basis if and only if this function is a uniform spline. This paper generalizes the above concept to pairs of spline functions \(\phi_1, \phi_2\) which are refinable in a vector sense, i.e., \(\phi=(\phi_1, \phi_2)^T\) is said to be \(m\)-refinable if \(\phi\) satisfies \[ \phi(x) = \sum_{j=-N}^{N} A(j) \phi(mx-j) \qquad (x \in {\mathbb{R}}) \] with \(2 \times 2\) coefficient matrices \(A(j)\). It is shown that refinability forces a special form of the Fourier transform of \(\phi_1, \phi_2\). Again an explicit characterization of \(\phi_1, \phi_2\) whose integer translates form a Riesz basis can be verified. The B-splines with double integer knots can be found as a special case of that characterization.
Mathematics Subject Classification ID: 41A15
Mathematics Subject Classification ID: 42C15
zbMATH DE Number: 1121282
zbMATH Keywords: refinable splines
zbMATH Keywords: matrix refinement equation
zbMATH Keywords: Riesz basis
reviewed by: Gerlind Plonka
MaRDI profile type: Publication
full work available at URL: jath.1996.3105
OpenAlex ID: W2076626479
cites work: Intertwining Multiresolution Analyses and the Construction of Piecewise-Polynomial Wavelets
cites work: Q4321676
cites work: Wavelets of Multiplicity r
cites work: Q3975831
cites work: On linear independence for integer translates of a finite number of functions
cites work: Characterization of compactly supported refinable splines
cites work: Two-scale symbol and autocorrelation symbol for B-splines with multiple knots
DOI: 10.1006/JATH.1996.3105
Recommended article: Q4378606
Recommended article: Characterization of compactly supported refinable splines
Recommended article: Q5451952
Recommended article: Q4542951
Recommended article: Structure of refinable splines
Recommended article: Q4938083
Recommended article: Classification of refinable splines
Recommended article: Q3839102
Recommended article: Q4240356
Recommended article: Classification of refinable splines in \(\mathbb R^{d }\)
description: scientific article; zbMATH DE number 1121282