x的有理插值的若干注记

摘要

The present paper constructs a set of nodes which can generate a rational interpolating function to approximate |x| at the rate of O(1/(nκlogn)) for any given natural number k. More importantly, this construction reveals the fact that the higher density the distribution of a set of nodes has to zero (that is the singular point of the function |x|!), the better the rational interpolation approximation behaves. This probably also provides an idea to construct more valuable sets of nodes in the future.%本文中我们构造了一个结点组,基于它定义的有理插值函数,对于任意给定的自然数κ,对|x|的逼近能达到精确阶O(1/(nklog n)).更重要的是,这样的构造揭示了一个本质:当结点向(|x|的唯一奇异点)零点集中时,|x|的有理插值逼近阶也随之更佳.这或许为将来本质性的自然结点组的构造提供了一种思路.