On an interpolation process of Lagrange-Hermite type

We consider a Lagrange-Hermite polynomial, interpolating a func- tion 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 L p -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.