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

Jean-Pierre Crouzeix (), Nadezda Sukhorukova () and Julien Ugon ()
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Jean-Pierre Crouzeix: Blaise Pascal University
Nadezda Sukhorukova: Swinburne University of Technology
Julien Ugon: Federation University Australia

Journal of Optimization Theory and Applications, 2017, vol. 172, issue 3, No 11, 950-964

Abstract: 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.

Keywords: Polynomial splines; Chebyshev approximation; Free knots; Quasidifferential; Best approximation conditions; 49J52; 90C26; 41A15; 41A50 (search for similar items in EconPapers)
Date: 2017
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Citations: View citations in EconPapers (1)

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DOI: 10.1007/s10957-016-1048-1

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