The paper is devoted to the study of a Pal type (0;1) interpolation problem on the unit circle considering two disjoint sets of nodes. The nodal points are obtained by projecting vertically the zeros of the Jacobi polynomial P _n^{(α,β)}(x) and its derivative P _n^{(α,β)'}(x) , together with ±1 onto the unit circle. The Lagrange data are prescribed on the first set of nodes, the Hermite data are prescribed on the second one and generalized Hermite-Fejer boundary conditions are prescribed at ±1. An explicit representation of the interpolatory polynomial is given and the convergence is studied for analytic functions on the unit disk. The results are of interest to approximation theory.

This research article aims to staunchly study the approximation using Lagrange interpolation on the unit circle. Nodal system constitutes the vertically projected zeros of Jacobi polynomial onto the unit circle with boundary points at ±1. Moreover, convergence is obtained by considering analytic functions on a suitable domain accompanied by some numerical experiments.

The paper is devoted to the study of a Pál type (0; 1) interpolation problem on the unit circle considering two disjoint sets of nodes. The nodal points are obtained by projecting vertically the zeros of the Jacobi polynomial P_n^(α, β)(x) and its derivative P_n^(α, β)' (x), together with ±1 onto the unit circle. The Lagrange data are prescribed on the first set of nodes, the Hermite data are prescribed on the second one and generalized Hermite-Fejér boundary conditions are prescribed at ±1. An explicit representation of the interpolatory polynomial is given and the convergence is studied for analytic functions on the unit disk. The results are of interest to approximation theory.

The aim of this paper is to study the approximation of functions using a higher order Hermite-Fejer interpolation process on the unit circle. The system of nodes is composed of vertically projected zeros of Jacobi polynomials onto the unit circle with boundary points at $ \pm1 $. Values of the polynomial and its first four derivatives are fixed by the interpolation conditions at the nodes. Convergence of the process is obtained for analytic functions on a suitable domain, and the rate of convergence is estimated.

In this paper we have constructed a non-interpolatory spline on the unit circle. The rate of convergence and the error in approximation corresponding to the complex valued function has been considered.