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 L p-spaces, 1 < p < ∞, 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.

Lagrange–Hermite interpolation on the real semiaxis

“…Therefore, in weighted spaces of continuous functions, the behaviour of the operators {L m,r (w)} m is comparable with similar interpolation processes based on Jacobi zeros on bounded intervals (see, e.g., [15,17]).…”