Best Approximation by Free Knot Splines

Meinardus, Günter ; Walz, Guido

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URN: urn:nbn:de:bsz:180-madoc-16060
Document Type: Working paper
Year of publication: 1998
The title of a journal, publication series: None
Publication language: English
Institution: School of Business Informatics and Mathematics > Sonstige - Fakultät für Wirtschaftsinformatik und Wirtschaftsmathematik
MADOC publication series: Veröffentlichungen der Fakultät für Mathematik und Informatik > Institut für Mathematik > Mannheimer Manuskripte
Subject: 510 Mathematics
Classification: MSC: 41A30 41A15 ,
Subject headings (SWD): Knotentheorie , Spline , Beste Approximation , Transformation <Mathematik>
Keywords (English): Free Knot Splines , Gauß transform , local Haar condition , nonlinear approximation , alternant theorems
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 Gauß 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.
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