Generalized Divided Differences, with Applications to Generalized B-Splines

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