Generalized Divided Differences, with Applications to Generalized B-Splines

Walz, Guido

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URN: urn:nbn:de:bsz:180-madoc-16782
Document Type: Working paper
Year of publication: 1992
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: 65D07 41A15 ,
Subject headings (SWD): B-Spline , Differenzenquotient , Rekursion
Keywords (English): Generalized B-Splines , Divided Differences , Recursion Schemes
Abstract: Within the theory of spline functions, there was always great interest in the study of B-splines, i.e. splines with a finite support. In addition to the classical polynomial case, there exist also various approaches to the definition of B-splines from other function classes, such as trigonometrie or hyperbolic ones (cf. section 0). In the present paper we define B-splines from a rat her general function space, which covers almost all existing approaches as special cases. The only condition that the spaces under consideration must satisfy is that of being translation invariant, a fundamental property which we are going to define in section 2. Our definition of generalized B-splines is based on generalized divided differences, which go back to Popoviciu [14] and were further investigated by Mühlbach [10, 11]. In the first section of this paper we therefore study these operators and prove new results, such as a contour integral representation and a multistep-formula; the latter one expresses - in closed form - a generalized divided difference of order m+j by those of order m, for arbitratry j ∈ IN. This makes it possible to compute generalized divided differences recursively, even if the underlying function space is spanned by a non-complete Chebyshev system.
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