Chebyshev--Schoenberg operators
type: Item
instance of: scholarly article
title: Chebyshev--Schoenberg operators
zbMATH Open document ID: 1229.41010
author: Marie-Laurence Mazure
published in: Constructive Approximation
publication date: 2011-09-27T00:00:00Z
review text: The present paper generalizes the results of \textit{M.-L. Mazure} [Numer. Algorithms 52, No. 1, 93--128 (2009; Zbl 1200.41023)] to Schoenberg-type operators. It is shown that a given spline space based on a given extended Chebyshev space gives birth to infinitely many variation-diminishing operators of Schoenberg-type, characterized by the two-dimensional Chebyshev spaces they reproduce. It is shown that the set of these two-dimensional spaces is stable under knot insertion and dimension elevation. The author considers the results of this article as originating in the powerful theory of Chebyshevian blossoms. Chebyshev-Schoenberg operators based on a given Chebyshevian spline space are defined. Building Chebyshev-Schoenberg operators is considered. Their shape preserving properties are discussed. A simple sufficient condition for convergence of infinite sequences of Chebyshev-Schoenberg operators is established. The relations between Chebyshev-Bernstein operators and Chebyshev-Schoenberg operators are considered. The obtained results are compared with those obtained by \textit{M. J. Marsden} [J. Approximation Theory 3, 7--49 (1970; Zbl 0192.42103)].
Mathematics Subject Classification ID: 41A15
Mathematics Subject Classification ID: 41A36
Mathematics Subject Classification ID: 41A50
Mathematics Subject Classification ID: 65D07
Mathematics Subject Classification ID: 65D17
zbMATH DE Number: 5951122
zbMATH Keywords: extended Chebyshev spaces
zbMATH Keywords: spline spaces
zbMATH Keywords: Schoenberg-type operators
zbMATH Keywords: Chebyshev-Bernstein operators
zbMATH Keywords: Chebyshev-Schoenberg operators
zbMATH Keywords: knot insertion
zbMATH Keywords: shape preservation
zbMATH Keywords: convergence
zbMATH Keywords: total positivity
zbMATH Keywords: blossoms
reviewed by: V. L. Leontiev
MaRDI profile type: Publication
full work available at URL: s00365-010-9123-6
OpenAlex ID: W2029776623
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DOI: 10.1007/S00365-010-9123-6
description: scientific article; zbMATH DE number 5951122