Piecewise Chebyshevian splines: interpolation versus design
type: Item
instance of: scholarly article
title: Piecewise Chebyshevian splines: interpolation versus design
zbMATH Open document ID: 1429.65034
author: Marie-Laurence Mazure
published in: Numerical Algorithms
publication date: 2018-04-16T00:00:00Z
review text: The author investigates the connections between Hermite interpolation and geometric design in spline spaces, inspired by an analogous result in the non-spline case of a general W-space on a non trivial interval $I$, i.e. a space of sufficiently differentiable functions where the Wronskian of any basis never vanishes on $I$. The most general spline Hermite interpolation problem is considered by introducing the W-splines, possibily taking into account the presence of connection matrices. Necessary conditions are provided for a given interpolation problem to be unisolvent, possibly splitting it into several simpler ones. Blossoms, generalised derivatives, Schoenberg-Whitney conditions are tools used in the proof of the main result of the paper that claims that (i) a piecewise W-spline space ${\mathbf S}$ good for interpolation is equivalent to (ii) the piecewise W-spline space ${\widehat{\mathbf S}}$ ``good for design'', obtained from the first one by continuous integration on $I$. Also, if (i) is satisfied, the space ${\widehat{\mathbf S}}$ is ``good for interpolation'' in turn. Such a problem is explored in details, by also proving that in a good for design piecewise W-spline space appropriate systems of weight functions can always be found so that, in terms of the associated generalised derivatives, all matrices involved in the connection conditions are identity matrices, the simplest possible totally positive matrices. Taking advantage from this facts, the unisolvence of a Hermite interpolation problem is characterised in terms of upper bounds on the number of zeros of splines in ${\mathbf S}$. Some examples are reported on Hermite interpolation with W-splines and on geometrically continuous cubic splines, showing explicit necessary and sufficient conditions for the spline space to be good for interpolation and for design. There are also comparisons that show the equivalence between these two facts and a specific interest in unisolvence of interpolation problems, w…
reviewed by: Paola Lamberti
Mathematics Subject Classification ID: 65D07
Mathematics Subject Classification ID: 41A15
Mathematics Subject Classification ID: 65D17
zbMATH DE Number: 6860409
zbMATH Keywords: piecewise Chebyshevian splines
zbMATH Keywords: connection matrices
zbMATH Keywords: spline Hermite interpolation
zbMATH Keywords: Schoenberg-Whitney conditions
zbMATH Keywords: total positivity
zbMATH Keywords: (piecewise) generalised derivatives
zbMATH Keywords: B-spline-type bases
zbMATH Keywords: knot insertion
zbMATH Keywords: blossoms
MaRDI profile type: Publication
full work available at URL: s11075-017-0360-7
OpenAlex ID: W2620964245
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DOI: 10.1007/S11075-017-0360-7
description: scientific article; zbMATH DE number 6860409