Some improvements to the Hermite–Fejér interpolation on the circle and bounded interval

In this paper, we study the convergence of the Hermite–Fejér and the Hermite interpolation polynomials, which are constructed by taking equally spaced nodes on the unit circle. The results that we obtain are concerned with the behaviour outside and inside the unit circle, when we consider analytic functions on a suitable domain. As a consequence, we achieve some improvements on Hermite interpolation problems on the real line. Since the Hermite–Fejér and the Hermite interpolation problems on [−1, 1], with nodal systems mainly based on sets of zeros of orthogonal polynomials, have been widely studied, in our contribution we develop a theory for three special nodal systems. They are constituted by the zeros of the Tchebychef polynomial of the second kind joint with the extremal points −1 and 1, the zeros of the Tchebychef polynomial of the fourth kind joint with the point −1, and the zeros of the Tchebychef polynomial of the third kind joint with the point 1. We present a simple and efficient method to compute these interpolation polynomials and we study the convergence properties.