Gibbs phenomenon in the Hermite interpolation on the circle

Hermite-Fejér interpolation problems on the unit circle and bounded interval are usually studied in relation with continuous functions. There are few references concerning these problems for functions with discontinuities. Thus the aim of this paper is to describe the behavior of the Hermite-Fejér and Hermite interpolants for piecewise continuous functions on the unit circle, analyzing the corresponding Gibbs phenomenon near the discontinuities and providing the asymptotic amplitude of the Gibbs height.