A note on optimal Hermite interpolation in Sobolev spaces

Abstract

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