样条函数变差缩减逼近法的迭代极限

本文是1]的进一步推广,即把1]中所考虑的三次等距节点的样条函数推广为任意(非等距)节点与任意幂次的多项式样条函数情形.对于最一般的多项式样条函数,我们证明了它的变差缩减逼近法当其迭代次数趋于无穷时也是收敛的,并且它的极限函数由折线多边形所组成(详见本文定理3).

ITERATED LIMIT FOR VARIATION DIMINISHING APPROXIMATION OF SPLINE FUNCTIONS

Let f(x) ∈Ca, b]. It is well known that Schoenberg-Marsden's variation-diminishing spline approximation is given by the followingLet S~k(f) (x) = S(S~(k-1)(f))(x), k > 1. We have the iterated esquence {S~k(f) (x)}. In this paper, we prove that S~k(f)(x) converge uniformly to F(x) as k→∞. Here, F(x) is a polygonal function connected by the points of (x_i),f(x_1), i = 1,…, r, for all knots of x's which are of multiplicity m-1 or m.Obviously, The iterated resalts of V. D. approximations such as those of Bernstein (Bezier) polynomial and simple (continuous) splines are all the special case of our general considerration.