Characterisation theorem for best polynomial spline approximation with free knots

Sukhorukova, Nadezda; Ugon, Julien

doi:10.1090/tran/6863

Characterisation theorem for best polynomial spline approximation with free knots
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by Nadezda Sukhorukova and Julien Ugon PDF

Trans. Amer. Math. Soc. 369 (2017), 6389-6405 Request permission

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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