B-splines im komplexen

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Abstract
In the present paper we study complex spline functions, which are defind as piecewise polynomial functions on a Jordan curve Г in the complex plane. Such splines we first introduced by Ahlberg and Nilson in 1965 [2] and further studied by Ahlberg., Nilson and Walsh [1], [3]-[5] and Schoenberg [9]. All these papers were mainly concerned with the interpolatory properties of complex splines. In contrast to the real case, there exist only very few results on complex B-splines; these are functions, which provide a numeically stable basis of the spline space S m.k . Although they are used in some papers (e.g. [6], [7], [9]), their main properties, for example the important recursion formula (3.8). were not proved until now. The reason for this might be that in the cited papers B-splines were defined as divided differences, applied to a certain truncated power function, a definition which is not at all easy to deal with. In this paper we present a more axiomatic approach to the B-splines B mv , which makes it possible to prove the main properties of these functions in a quite elementary way; this approach is inspired in some parts by recent results on real B-splines, due to G. Meinardus [8]. As it turns out, one needs nothing but the minimal-support property of B-splines in order to prove that they form a basis of the spline space. In addition, we show that our definition leads essentially to the same functions as in [6], [7], and, furthermore that some other properties of real B-splines carry over to the complex case. For example, there exists a represention of B mv via a complex contour integral.

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