Optimal Lagrange Interpolation of a Class of Infinitely Differentiable Functions
Received:November 09, 2020  Revised:January 03, 2021
Key Words: worst case setting   optimal Lagrange interpolation   infinitely differentiable function space
Fund Project:Supported by the National Natural Science Foundation of China (Grant No.11871006).
 Author Name Affiliation Mengjin MA Department of Mathematics, Tianjin Normal University, Tianjin 300387, P. R. China Hui WANG Department of Mathematics, Tianjin Normal University, Tianjin 300387, P. R. China Guiqiao XU Department of Mathematics, Tianjin Normal University, Tianjin 300387, P. R. China
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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.