Convergence of Hermite interpolants on the circle using two derivatives

In this paper we deal with Hermite interpolation problems on the unit circle considering up to the second derivative for the interpolation conditions and taking equally spaced points as nodal system. In the extended Fejér case, which corresponds to take vanishing values for the first two derivatives, we prove the uniform convergence for the interpolants related to continuous functions with smooth modulus of continuity. We also consider the Hermite case with non vanishing conditions for the derivatives for which we establish sufficient conditions on the interpolation conditions to obtain convergence.