Tschebyscheff-approximation mit rationalen splines bei freien knoten

Abstract

A function s ϵ C n [ α, β] is called a rational spline, if s ( n) is positive, and if there exist knots α = x 0 < x 1 < … < x m + 1 = β such that the restrictions s¦[x i, x i + 1] are rational functions with nth-degree numerator and linear denominator. Let S nk [ α, β] be the set of rational splines having at most k knots, and let S nk[α, β] be its closure with respect to the topology of uniform convergence. It is shown that, for each ƒ ϵ C[α, β], there exists a best approximation s ∗ ϵ S nk[α, β] (in the Chebyshev sense). The splines belonging to S nk[α, β] nk[α, β] are characterized explicitly: They consist of rational functions, too, but have multiple knots, the number of which is bounded by k, if counted multiply. However, in contrast to polynomial splines, at most threefold knots occur. Finally, some results on the existence of best smooth approximations in case n = 2 are reported.