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Characterization Theorem for Best Polynomial Spline Approximation with Free Knots, Variable Degree and Fixed Tails

Author

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  • Jean-Pierre Crouzeix

    (Blaise Pascal University)

  • Nadezda Sukhorukova

    (Swinburne University of Technology
    Federation University Australia)

  • Julien Ugon

    (Federation University Australia)

Abstract

In this paper, we derive a necessary condition for a best approximation by piecewise polynomial functions of varying degree from one interval to another. Based on these results, we obtain a characterization theorem for the polynomial splines with fixed tails, that is the value of the spline is fixed in one or more knots (external or internal). We apply nonsmooth nonconvex analysis to obtain this result, which is also a necessary and sufficient condition for inf-stationarity in the sense of Demyanov–Rubinov. This paper is an extension of a paper where similar conditions were obtained for free tails splines. The main results of this paper are essential for the development of a Remez-type algorithm for free knot spline approximation.

Suggested Citation

  • Jean-Pierre Crouzeix & Nadezda Sukhorukova & Julien Ugon, 2017. "Characterization Theorem for Best Polynomial Spline Approximation with Free Knots, Variable Degree and Fixed Tails," Journal of Optimization Theory and Applications, Springer, vol. 172(3), pages 950-964, March.
    • Handle: RePEc:spr:joptap:v:172:y:2017:i:3:d:10.1007_s10957-016-1048-1
    DOI: 10.1007/s10957-016-1048-1
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    References listed on IDEAS

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    1. Adil Bagirov & Napsu Karmitsa & Marko M. Mäkelä, 2014. "Introduction to Nonsmooth Optimization," Springer Books, Springer, edition 127, number 978-3-319-08114-4, February.
    2. Nadezda Sukhorukova, 2010. "Uniform Approximation by the Highest Defect Continuous Polynomial Splines: Necessary and Sufficient Optimality Conditions and Their Generalisations," Journal of Optimization Theory and Applications, Springer, vol. 147(2), pages 378-394, November.
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    Cited by:

    1. Peiris, V. & Sharon, N. & Sukhorukova, N. & Ugon, J., 2021. "Generalised rational approximation and its application to improve deep learning classifiers," Applied Mathematics and Computation, Elsevier, vol. 389(C).

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