Optimal Lagrange interpolation of a class of infinitely differentiable functions
Received:September 08, 2020  Revised:December 05, 2020
Key Word: worst case setting   optimal Lagrange interpolation polynomial   infinitely differentiable function space.
Fund ProjectL:National Natural Science Foundation of China (11871006),
 Author Name Affiliation Postcode Mengjin Ma Tianjin Normal University, College of Mathematical Science 300387 Guiqiao Xu Tianjin Normal University, College of Mathematical Science Hui Wang Tianjin Normal University, College of Mathematical Science 300387
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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 prove 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 we ask the endpoints to be included in the nodes.