Characterization theorem for best polynomial spline approximation with free knots

TL;DR

This paper establishes a necessary and sufficient condition for optimal polynomial spline approximation with free knots, using nonsmooth nonconvex analysis to improve upon existing characterizations.

Contribution

It introduces a new characterization theorem for best free knots polynomial spline approximation, strengthening previous results by incorporating a special property of the knots.

Findings

Derived a necessary condition for best approximation by free knot splines

Applied nonsmooth nonconvex analysis to obtain the characterization

Provided a stronger characterization than previous results

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 characterization theorem for best free knots polynomial spline approximation, which is stronger than the existing characterisation results when only continuity is required.