Sample numbers and optimal Lagrange interpolation in Sobolev spaces

Abstract

This paper investigates the optimal recovery of Sobolev spaces $W_{\infty }^r\left[-1,1\right],r\in \mathbb{N}$ in space $L_{\infty }\left[-1,1\right]$ and weighted spaces $L_{p,\omega }\left[-1,1\right],1\le p<\infty$ with $\omega$ a continuous integrable weight function in $\left(-1,1\right)$. We obtain the values of the sampling numbers of $W_{\infty }^r\left[-1,1\right]$ in $L_{\infty }\left[-1,1\right]$ and $L_{p,\omega }\left[-1,1\right],1\le p<\infty$. We prove that the Lagrange interpolation algorithms based on the Chebyshev nodes of the first kind are optimal for $p=\infty$. Meanwhile, we prove that the Lagrange interpolation algorithms based on the zeros of polynomial of degree $r$ with the leading coefficient $1$ of the least deviation from zero in $L_{p,\omega }\left[-1,1\right]$ are optimal for $1\le p<\infty$. We also give the optimal Lagrange interpolation algorithms when we ask the endpoints to be included in the nodes.