The metric projection for free knot splines

Nürnberger, Günther

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URN: urn:nbn:de:bsz:180-madoc-16888
Document Type: Working paper
Year of publication: 1990
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: 41A15 41A50 41A65 ,
Subject headings (SWD): Knotentheorie , Spline , Gleichmäßige Approximation , Metrische Projektion
Keywords (English): best uniform approximation , metric projection
Abstract: Only few results are known on continuity properties of the set-valued metric projection in nonlinear uniform approximation. In this paper we investigate this mapping in the case of best uniform approximation by splines of degree m with k free knots. A characterization of those functions at which the metric projection is upper semicontinuous is given. It follows that the metric projection is upper semicontinuous if and only if k ≤ m, and that it is upper semicontinuous at all "normal" functions. On the other hand, it is shown that the metric projection is never lower semicontinuous.
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