On a new criterion to decide whether a spline space can be used for design
type: Item
instance of: scholarly article
title: On a new criterion to decide whether a spline space can be used for design
zbMATH Open document ID: 1259.65020
author: Marie-Laurence Mazure
published in: BIT
publication date: 2012-12-20T00:00:00Z
review text: Using the most suitable approach of blossoming, the author considers the question of usefulness of piecewise approximations taken from extended Chebyshev spaces for design purposes. The idea is to take at first piecewise functions of the spline type (classically these would be piecewise polynomial functions), each piece being from an extended (generalised) Chebyshev space instead of just a polynomial space of degree \(n\), say. Theses pieces are linked to each other under the conditions of smoothness of some order at the knots, where in this situation the smoothness is considered by conditions on the parameters. In this generality, it is not evident which are the cases where the resulting functions are suitable for design and which are not. It is the purpose of the paper to find well-applicable conditions for that question and this is nicely achieved by the tool of blossoming. The paper provides a good overview of the framework of the approach at the beginning. The criterion is provided in Section~3. Useful, illuminating examples are given for the case \(n=3\) (in the classical polynomial case that is the degree of the piecewise polynomials) and the special case of zero-multiplicity at the knots. That and some further remarks on some other important special cases end this interesting work.
reviewed by: Martin D. Buhmann
Mathematics Subject Classification ID: 65D07
Mathematics Subject Classification ID: 41A15
zbMATH DE Number: 6117474
zbMATH Keywords: geometric design
zbMATH Keywords: extended Chebyshev spaces
zbMATH Keywords: Bernstein bases
zbMATH Keywords: weight functions
zbMATH Keywords: generalised derivatives
zbMATH Keywords: B-spline bases
zbMATH Keywords: total positivity
zbMATH Keywords: blossoms
zbMATH Keywords: numerical examples
MaRDI profile type: Publication
full work available at URL: s10543-012-0390-4
OpenAlex ID: W2032145396
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DOI: 10.1007/S10543-012-0390-4
description: scientific article; zbMATH DE number 6117474