Optimal Lagrange Interpolation of a Class of Infinitely Differentiable Functions

Abstract:

      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.