连续可导函数类的最优拉格朗日插值  

OPTIMAL LAGRANGE INTERPOLATION FOR CONTINUOUSLY DIFFERENTIABLE FUNCTIONS

作　　者：齐宗会[1] 汪晖 刘永平[3] Qi Zonghui;Wang Hui;Liu Yongping(Boustead College,TianJin Commerce University,Tianjin 300384;Department of Mathematics,Tianjin Normal University,Tianjin 300387)

机构地区：[1]天津商业大学宝德学院,天津300384 [2]天津师范大学数学科学学院,天津300387 [3]北京师范大学数学科学学院,北京100875

出　　处：《高等学校计算数学学报》2020年第1期87-96,共10页Numerical Mathematics A Journal of Chinese Universities

基　　金：国家自然科学基金支持项目(项目编号No.11871006).

摘　　要：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.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.

关 键 词：拉格朗日插值 插值结点 可导函数 插值函数 等距结点 龙格现象 无穷大 逼近 

分 类 号：O174.41[理学—数学]