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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.doi:10.1090/s0025-5718-2012-02645-7 fatcat:5mpebyk4fzhrnhn7c6xfnd4b5i