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Design or not design? A numerical characterisation for piecewise Chebyshevian splines

Authors
  • Beccari, Carolina Vittoria1
  • Casciola, Giulio1
  • Mazure, Marie-Laurence2
  • 1 University of Bologna, Department of Mathematics, P.zza di Porta S. Donato 5, Bologna, 40126, Italy , Bologna (Italy)
  • 2 Université Grenoble-Alpes, Laboratoire Jean Kuntzmann, BP53, Cedex 9, Grenoble, 38041, France , Grenoble (France)
Type
Published Article
Journal
Numerical Algorithms
Publisher
Springer US
Publication Date
May 12, 2018
Volume
81
Issue
1
Pages
1–31
Identifiers
DOI: 10.1007/s11075-018-0533-z
Source
Springer Nature
Keywords
License
Yellow

Abstract

This work concerns the useful and large class of all piecewise Chebyshevian splines, in the sense of splines with pieces taken from different Extended Chebyshev spaces all of the same dimension, and with connection matrices at the knots. The subclass of those which are interesting for applications, and in particular for design, is known to be characterised by the fact that the continuity between consecutive pieces can always be controlled by identity matrices, provided that we express it by means of appropriate generalised derivatives associated with the section-spaces. Modelled on the proof of this beautiful theoretical characterisation, we provide a numerical procedure to check whether or not a given spline space lies in that subclass. Examples are given proving the usefulness of the test in situations where it is not expectable to derive exact practical conditions from the above-mentioned theoretical characterisation.

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