On an interpolation process of Lagrange-Hermite type

Abstract

We consider a Lagrange-Hermite polynomial, interpolating a function at the Jacobi zeros and, with its first (r?1) derivatives, at the points ?1. We give necessary and sufficient conditions on the weights for the uniform boundedness of the related operator in certain suitable weighted Lp-spaces, 1 < p < 1, proving a Marcinkiewicz inequality involving the derivative of the polynomial at ?1. Moreover, we give optimal estimates for the error of this process also in the weighted uniform metric.