Gibbs–Wilbraham oscillation related to an Hermite interpolation problem on the unit circle

Abstract

The aim of this piece of work is to study some topics related to an Hermite interpolation problem on the unit circle. We consider as nodal points the zeros of the para-orthogonal polynomials with respect to a measure in the Baxter class and such that the sequence of the first derivative of the reciprocal of the orthogonal polynomials is uniformly bounded on the unit circle. We study the convergence of the Hermite–Fejér interpolants related to piecewise continuous functions and we describe the sets in which the interpolants uniformly converge to the piecewise continuous function as well as the oscillatory behavior of the interpolants near the discontinuities, where a Gibbs–Wilbraham phenomenon appears. Finally we present some numerical experiments applying the main results and by considering nodal systems of interest in the theory of orthogonal polynomials.