Higher order Hermite-Fejer Interpolation on the unit circle

TL;DR

This paper investigates higher order Hermite-Fejer interpolation on the unit circle, focusing on convergence and rate estimates for analytic functions using Jacobi polynomial zeros as nodes.

Contribution

It introduces a novel interpolation process on the unit circle with fixed derivatives at Jacobi polynomial zeros, providing convergence analysis and rate estimates.

Findings

Convergence established for analytic functions.

Rate of convergence estimated.

Interpolation process effective on the unit circle.

Abstract

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.