连续可导函数类的最优拉格朗日插值

在构造拉格朗日插值算法时,插值结点的选择是十分重要的.给定一个足够光滑的函数,如果结点选择的不好,当插值结点个数趋于无穷时,插值函数不收敛于函数本身.例如龙格现象:对于龙格函数f(x)=1/1+25x^2,如果拉格朗日插值的结点取-1,1]上的等距结点,那么逼近的误差会随着结点个数增多而趋于无穷大⑴,由此可知插值结点的选择尤为重要.

OPTIMAL LAGRANGE INTERPOLATION FOR CONTINUOUSLY DIFFERENTIABLE FUNCTIONS

Let C^n(-1,1])denote the space of the functions defined on the interval-1,1]and having continuous derivatives up to n^th order.In the approximation problem by Lagrange interpolation polynomials,it is well known that the optimal n-1 Lagrange interpolation nodes for C^n(-1,1])under the uniform norm L∞ are the zeros of n^th Chebyshev polynomial.Recently,N.S.Hoang gave the optimal n-1 Lagrange interpolation nodes for C^n(-1,1])under the uniform norm L∞ when the interval endpoints are also included in the interpolation node set.In this paper,we give the optimal n-1 Lagrange interpolation nodes for C^n(-1,1])under the norm Lp(1≤p<∞),and the optimal n-1 Lagrange interpolation nodes for C(-1,1])under the norm Lp(1≤p<∞)when the interval endpoints are also included in the interpolation node set.In addition,we corrected the optimal deviation given in a paper of Babave and Hoyotov.