On Exponential Convergence of Gegenbauer Interpolation and Spectral Differentiation

TL;DR

This paper provides a rigorous analysis of the exponential convergence rates of Gegenbauer polynomial interpolation and spectral differentiation for analytic functions, with sharp error estimates in the maximum norm.

Contribution

It offers the first rigorous proof of exponential convergence for Gegenbauer-based spectral methods with precise error bounds.

Findings

Exponential convergence rates are established for Gegenbauer interpolation.

Sharp error estimates in the maximum norm are derived.

Results apply to functions analytic within an ellipse.

Abstract

This paper is devoted to a rigorous analysis of exponential convergence of polynomial interpolation and spectral differentiation based on the Gegenbauer-Gauss and Gegenbauer-Gauss-Lobatto points, when the underlying function is analytic on and within an ellipse. Sharp error estimates in the maximum norm are derived.