Sharp error bounds for Jacobi expansions and Gengenbauer-Gauss quadrature of analytic functions

TL;DR

This paper derives sharp exponential decay bounds for Jacobi polynomial expansions and Gegenbauer-Gauss quadrature errors of analytic functions, with explicit parameter dependence, improving on existing estimates.

Contribution

It introduces new, explicit bounds for Jacobi expansion coefficients and quadrature errors, extending previous results with sharper estimates and detailed parameter dependence.

Findings

Derived sharp exponential decay bounds for Jacobi expansions.

Established new tight bounds for Gegenbauer-Gauss quadrature errors.

Validated bounds by comparison with recent existing results.

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 [38, SIAM J. Numer. Anal., 2012]. 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.