Some Identities for Trigonometric B-Splines, with an Application to Curve Design

Walz, Guido

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URN: urn:nbn:de:bsz:180-madoc-17112
Document Type: Working paper
Year of publication: 1995
Publication language: English
Institution: School of Business Informatics and Mathematics > Sonstige - Fakultät für Mathematik und Informatik
MADOC publication series: Veröffentlichungen der Fakultät für Mathematik und Informatik > Institut für Mathematik > Mannheimer Manuskripte
Subject: 510 Mathematics
Classification: MSC: 42A10 41A15 ,
Subject headings (SWD): Rekursionsformel , Trigonometrie , B-Spline , Integraldarstellung , Konvexe Hülle
Keywords (English): Trigonometric Splines , Trigonometric B-Splines , Partition of Unity , Convex-Hull Property , Integral Representation , Recursion Formula
Abstract: In this paper we investigate some properties of trigonometric B-splines, which form a finitely-supported basis of the space of trigonometric spline functions. We establish a complex integral representation for trigonometric B-splines, which is in certain analogy to the polynomial case, but the proof of which has to be done in a different and more complicated way. Using this integral representation, we can prove some identities concerning the evaluation of a trigonometric B-spline, its derivative and its partial derivative w.r.t. the knots. As a corollary of the last mentioned identity, we obtain a result on the tangent space of a trigonometric spline function. Finally we show that - in the case of equidistant knots - the trigonometric B-splines of odd order form a partition of a constant, and therefore the corresponding B-spline curve possesses the convex-hull property. This is also illustrated by a numerical example.
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