Find in Library

Abstract This paper investigates the optimal Hermite interpolation of Sobolev spaces W ∞ n [ a , b ] $W_{\infty }^{n}[a,b]$ , n ∈ N $n\in \mathbb{N}$ in space L ∞ [ a , b ] $L_{\infty }[a,b]$ and weighted spaces L p , ω [ a , b ] $L_{p,\omega }[a,b]$ , 1 ≤ p < ∞ $1\le p< \infty $ with ω a continuous-integrable weight function in ( a , b ) $(a,b)$ when the amount of Hermite data is n. We proved that the Lagrange interpolation algorithms based on the zeros of polynomial of degree n with the leading coefficient 1 of the least deviation from zero in L ∞ $L_{\infty }$ (or L p , ω [ a , b ] $L_{p,\omega }[a,b]$ , 1 ≤ p < ∞ $1\le p<\infty $ ) are optimal for W ∞ n [ a , b ] $W_{\infty }^{n}[a,b]$ in L ∞ [ a , b ] $L_{\infty }[a,b]$ (or L p , ω [ a , b ] $L_{p,\omega }[a,b]$ , 1 ≤ p < ∞ $1\le p<\infty $ ). We also give the optimal Hermite interpolation algorithms when we assume the endpoints are included in the interpolation systems.