Orthonormality of cardinal Chebyshev B-spline bases in weighted Sobolev spaces
type: Item
instance of: scholarly article
title: Orthonormality of cardinal Chebyshev B-spline bases in weighted Sobolev spaces
zbMATH Open document ID: 1020.65012
author: Marie-Laurence Mazure
author: Khaled Melkémi
published in: Constructive Approximation
publication date: 2003-09-18T00:00:00Z
review text: There is a celebrated result of \textit{U. Reif} [SIAM J. Math. Anal. 28, No.~5, 1258-1263 (1997; Zbl 0884.41009)] which shows that the classical polynomial (univariate) B-splines satisfy an orthonormality property with respect to a weighted Sobolev-inner product. This makes them even more useful for least-squares approximation which is facilitated by the provision of orthogonal bases. In this article, the authors generalise the theorem to Chebyshev B-splines which emerge from certain differential operators, closely related to the L-splines by \textit{C. A. Micchelli} [Stud. Spline Funct. Approx. Theory, 203-250 (1976; Zbl 0343.41008)]. The differential operators are expressed as products of shifts of derivative operators, and both cases when the shifts are either real or complex are studied. The B-splines themselves are differences of Green functions of the said differential operators.
Mathematics Subject Classification ID: 65D07
Mathematics Subject Classification ID: 41A15
Mathematics Subject Classification ID: 41A50
zbMATH DE Number: 1885727
zbMATH Keywords: Chebyshev B-splines
zbMATH Keywords: weighted Sobolev splines
zbMATH Keywords: weighted Sobolev spaces
zbMATH Keywords: orthonormality property
zbMATH Keywords: least-squares approximation
reviewed by: Martin D. Buhmann
MaRDI profile type: Publication
full work available at URL: s00365-001-0023-7
OpenAlex ID: W372196690
DOI: 10.1007/S00365-001-0023-7
description: scientific article; zbMATH DE number 1885727