Best Approximation by Free Knot Splines

Abstract

We consider the problem of finding the best (uniform) approximation of a given continuous function by spline functions with free knots. Our approach can be sketched as follows: By using the Gaus transform with arbitrary positive real parameter t, we map the set of splines under consideration onto a function space, which is arbitrarily elose to the spline set, but satisfies the local Haar condition and also possesses other nice structural properties. This enables us to give necessary and sufficient conditions for best approximations (in terms of alternants) and, under some assumptions, even full characterizations and a uniqueness result. By letting t -> 0, we recover a best approximation in the original spline space. Our results are illustrated by some numerical examples, which show in particular the nice alternation behavior of the error function.