Optimal Lagrange Interpolation of a Class of Infinitely Differentiable Functions

DOI：10.3770/j.issn:2095-2651.2021.06.007

 作者 单位 马孟瑾 天津师范大学数学学院, 天津 300387 汪珲 天津师范大学数学学院, 天津 300387 许贵桥 天津师范大学数学学院, 天津 300387

本文研究\,$[-1,1]$上的一个无限可微函数类$F_\infty$在空间$L_\infty[-1,1]$及加权空间$L_{p,\omega}[-1,1]$, $1\le p< \infty$ ($\omega$是$(-1,1)$上的非负连续可积函数)的最优Lagrange插值.我们证明了基于首项系数为1且于$L_{p,\omega}[-1,1]$上有最小范数的多项式零点的Lagrange插值对$1\le p< \infty$是最优的. 同时我们给出了当结点组包含端点时的最优结点组.

This paper investigates the optimal Lagrange interpolation of a class $F_\infty$ of infinitely differentiable functions on $[-1,1]$ in $L_\infty[-1,1]$ and weighted spaces $L_{p,\omega}[-1,1], \ 1\le p< \infty$ with $\omega$ a continuous integrable weight function in $(-1,1)$. We proved that the Lagrange interpolation polynomials based on the zeros of polynomials with the leading coefficient $1$ of the least deviation from zero in $L_{p,\omega}[-1,1]$ are optimal for $1\le p<\infty$. We also give the optimal Lagrange interpolation nodes when the endpoints are included in the nodes.