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Transactions of the American Mathematical Society

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Characterisation theorem for best polynomial spline approximation with free knots

Authors: Nadezda Sukhorukova and Julien Ugon
Journal: Trans. Amer. Math. Soc. 369 (2017), 6389-6405
MSC (2010): Primary 49J52, 90C26, 41A15, 41A50
Published electronically: March 17, 2017
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Abstract: In this paper, we derive a necessary condition for a best approximation by piecewise polynomial functions. 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. We start from identifying a special property of the knots. Then, using this property, we construct a characterisation theorem for best free-knots polynomial spline approximation, which is stronger than the existing characterisation results, at least in the case when only continuity is required.

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Additional Information

Nadezda Sukhorukova
Affiliation: Swinburne University of Technology, P.O. Box 218, Hawthorn, Victoria 3122, Australia
Address at time of publication: Faculty of Science, Federation University, P.O. Box 663, Ballarat, Victoria 3353, Australia

Julien Ugon
Affiliation: Centre for Informatics and Applied Optimization, Federation University, P.O. Box 663, Ballarat, Victoria 3353, Australia

Received by editor(s): September 17, 2013
Received by editor(s) in revised form: September 18, 2013, and September 27, 2015
Published electronically: March 17, 2017
Article copyright: © Copyright 2017 American Mathematical Society

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