Algorithms, Convergence and Rate of Convergence for an Interpolation Model Between Lagrange and Hermite

The aim of this piece of work is to study an interpolation problem on the interval $[-1,1]$, which can be considered an intermediate case between the interpolation problems of Lagrange and Hermite. The nodal points belong to the Chebyshev–Lobatto nodal system and the novelty is that we use the complete system for the Lagrange data and half of the nodes for the derivative data. When the extremal points are not used for the values of the derivative we have a quasi-interpolation problem. First we give two different types of algorithms for computing the interpolation polynomials. One of the expressions is given in terms of the Chebyshev basis of the first kind and the second one is based on a barycentric formula. The second part of the paper is devoted to obtain some results about the convergence and the rate of convergence of the interpolants when interpolating some type of smooth functions. We also consider the case of merely continuous functions obtaining a result of convergence which is closer to the Lagrange problem. Finally, we analyze the quasi-interpolation case.