On rational B-splines with prescribed poles
type: Item
instance of: scholarly article
title: On rational B-splines with prescribed poles
zbMATH Open document ID: 1063.41003
author: B. Buchwald
author: Günter W. Mühlbach
published in: Journal of Computational and Applied Mathematics
publication date: 2004-08-10T00:00:00Z
review text: Let \([\alpha,\beta]\) a real interval and \(P_2,P_3,\dots,P_m \notin [\alpha,\beta]\) (\(P_1\) is considered \(\infty\)). The authors of this paper consider Cauchy-Vandermonde systems (CV systems) \[ (1,x,\dots,x^{n_1-1},{{1}\over {x-P_2}},\dots,{{1}\over {(x-P_2)^{n_2}}},\dots,{{1}\over {x-P_m}},\dots,{{1}\over {(x-P_m)^{n_m}}}) \] giving rise to rational functions with prescribed poles. Here \(n_1\geq 1, \,n_i\geq 0 \quad \forall i>1\). The authors show how to construct an extended complete Chebyshev system (ECT system) in canonical form from the CV system above. Then they construct a space of rational splines with prescribed poles for a sequence of simple knots, and a basis of B-splines formed by splines with minimal compact support normalized to have integral 1 along the real line. All the constructions depend on a parameter \(\epsilon>0\) because the CV system used in the interval beteeen consecutive knots \([xi,x_{i+1}]\) is \[ (1,x-x_i,\dots,(x-x_i)^{d-3},{{1}\over {x-x_i+\epsilon}},{{1}\over {x-x_{i+1}-\epsilon}}). \] In the main result the authors prove that in the limit \(\epsilon \to 0\) the rational B-splines tend to certain polynomial B-splines.
reviewed by: M. Gasca
Mathematics Subject Classification ID: 41A15
Mathematics Subject Classification ID: 41A05
Mathematics Subject Classification ID: 41A20
zbMATH DE Number: 2085542
zbMATH Keywords: Cauchy-Vandermonde systems
zbMATH Keywords: rational B-splines
zbMATH Keywords: prescribed poles
MaRDI profile type: Publication
full work available at URL: j.cam.2003.10.004
OpenAlex ID: W2001424242
cites work: Construction of B-splines for generalized spline spaces generated from local ECT-systems
cites work: The Neville-Aitken formula for rational interpolants with prescribed poles
cites work: Computation of rational interpolants with prescribed poles
cites work: Fast Gaussian Elimination with Partial Pivoting for Matrices with Displacement Structure
cites work: Rational B-splines with prescribed poles
cites work: Q5534404
cites work: A recurrence relation for Chebyshevian B-splines
cites work: Q3922071
cites work: Interpolation by Cauchy-Vandermonde systems and applications
cites work: Q3895980
cites work: Displacement structure approach to Cauchy and Cauchy-Vandermonde matrices: Inversion formulas and fast algorithms
DOI: 10.1016/J.CAM.2003.10.004
Recommended article: Rational B-splines with prescribed poles
Recommended article: Approximating uniform rational B-spline curves by polynomial B-spline curves
Recommended article: Q5750325
Recommended article: Q3839095
Recommended article: Q4252676
Recommended article: A recurrence relation for rational B-splines
Recommended article: Q3582608
Recommended article: On Some Generalizations of B-Splines
Recommended article: Q3346745
Recommended article: Q5476404
description: scientific article; zbMATH DE number 2085542