Sharp error bounds for Jacobi expansions and Gengenbauer-Gauss quadrature of analytic functions
Provides rigorous, parameter-explicit error bounds for Jacobi expansions and Gegenbauer-Gauss quadrature, improving upon existing results for numerical analysts working on spectral methods and quadrature.
This paper derives sharp, explicit bounds for the exponential decay of Jacobi polynomial expansion coefficients of analytic functions, recovering the best known Chebyshev estimates, and extends these bounds to Gegenbauer-Gauss quadrature errors. The bounds are shown to be tight by comparison with recent results.
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.