Piecewise Chebyshev-Schoenberg operators: shape preservation, approximation and space embedding
type: Item
instance of: scholarly article
title: Piecewise Chebyshev-Schoenberg operators: shape preservation, approximation and space embedding
zbMATH Open document ID: 1272.41012
author: Marie-Laurence Mazure
published in: Journal of Approximation Theory
publication date: 2013-02-19T00:00:00Z
review text: The aim of the paper is to discuss the possibility of the construction of an analogue of the Schoenberg operator and to study its properties in the following very general framework. Consider the points \(a=t_0<t_1<\dotsb<t_q<t_{q+1}=b\) and associate with the interior points the multiplicities \(m_1,\dotsc,m_q\), with \(0\leq m_k\leq n\), for \(1\leq k\leq q\). Let there be given a sequence of spaces \(\mathbb{E}_k\subset C^n([t_k,t_{k+1}])\), \(0\leq k\leq q\), containing constants such that \(D\mathbb{E}_k:=\{DF:=F'\mid F\in \mathbb{E}_k\}\) are \(n\)-dimensional extended Chebyshev spaces. Suppose also that there is a sequence \(M_k\), \(1\leq k\leq q\) of connection matrices, each \(M_k\) being a lower triangular matrix of order \(n-m_k\) with positive diagonal entries. In this context consider the spline space \(\mathbb{S}\), of continuous functions \(S:[a,b]\to\mathbb{R}\) for which the restrictions to \([t_k,t_{k+1}]\) belong to \(\mathbb{E}_k\) for \(0\leq k\leq q\) and which satisfy the connection conditions \[ (DS(t_k^+),\dotsc,D^{n-m_k}S(t_k^+))= M_k(DS(t_k^-),\dotsc,D^{n-m_k}S(t_k^-)),\quad\text{ for }1\leq k\leq q. \] The construction of Schoenberg operators in the particular case where the section-spaces \(\mathbb{E}_k\) are obtained by restricting to \([t_k,t_{k+1}]\) of some space \(\mathbb{E}\subset C^n([a,b])\) was analysed in a previous paper by the author [Constr. Approx. 34, No. 2, 181--208 (2011; Zbl 1229.41010)]. In the generalized framework, the requirements imposed on the sequence \(\mathbb{E}_k\), \(0\leq k\leq q\), assure the existence of the blossom function, which plays a crucial role in the existence of \(B\)-spline basis, [the author, Constr. Approx. 20, No. 4, 603--624 (2004; Zbl 1063.65017)]. A piecewise Chebyshev-Schoenberg operator \(\mathcal{S}\) on \(C([a,b])\) is of the form \(\mathcal{S}F=\sum_{l=-n}^{m}F(\eta_l)N_l\), \(F\in C([a,b])\), where \(N_l\) is a \(B\)-spline basis on the knots \(\eta_l\), \(-n\leq l\leq m\) and \(\mat…
reviewed by: Radu Paltanea
Mathematics Subject Classification ID: 41A35
Mathematics Subject Classification ID: 41A15
Mathematics Subject Classification ID: 41A50
zbMATH DE Number: 6137310
zbMATH Keywords: piecewise-Chebyshev splines
zbMATH Keywords: connection matrices
zbMATH Keywords: \(B\)-spline bases
zbMATH Keywords: knots insertion
zbMATH Keywords: dimension elevation
zbMATH Keywords: blossoms
zbMATH Keywords: variation diminishing
zbMATH Keywords: Schoenberg-type operators
zbMATH Keywords: simultaneous approximation
MaRDI profile type: Publication
full work available at URL: j.jat.2012.10.005
OpenAlex ID: W1984756132
DOI: 10.1016/J.JAT.2012.10.005
description: scientific article; zbMATH DE number 6137310