A note on optimal Hermite interpolation in Sobolev spaces

This paper investigates the optimal Hermite interpolation of Sobolev spaces W-infinity(n)[a, b], n is an element of N in space L-infinity[a, b] and weighted spaces L-p,L-omega[a, b], 1 <= p < infinity with omega a continuous-integrable weight function in (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-infinity (or L-p,L-omega[a, b], 1 <= p < infinity) are optimal for W-infinity(n)[a, b] in L-infinity[a, b] (or L-p,L-omega[a, b], 1 <= p < infinity). We also give the optimal Hermite interpolation algorithms when we assume the endpoints are included in the interpolation systems.