Sharp Error Bounds for Jacobi Expansions and Gegenbauer--Gauss Quadrature of Analytic Functions

Abstract

This paper provides a rigorous and delicate analysis for exponential decay of Jacobi polynomial expansions of analytic functions associated with the Bernstein ellipse. Using an argument that can recover the best estimate for the Chebyshev expansion, we derive various new and sharp bounds of the expansion coefficients, which are featured with explicit dependence of all related parameters and valid for degree $n\ge 1$. We demonstrate the sharpness of the estimates by comparing with existing ones, in particular, the very recent results in SIAM J. Numer. Anal., 50 (2012), pp. 1240--1263. We also extend this argument to estimate the Gegenbauer--Gauss quadrature remainder of analytic functions, which leads to some new tight bounds for quadrature errors.