Sharp error bounds for Jacobi expansions and Gegenbauer--Gauss quadrature of analytic functions
Date of Issue2013
School of Physical and Mathematical Sciences
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.
DRNTU::Science::Mathematics::Applied mathematics::Numerical analysis
SIAM Journal on Numerical Analysis
© 2013 Society for Industrial and Applied Mathematics. This paper was published in SIAM Journal on Numerical Analysis and is made available as an electronic reprint (preprint) with permission of Society for Industrial and Applied Mathematics. The paper can be found at the following official DOI: http://dx.doi.org/10.1137/12089421X. One print or electronic copy may be made for personal use only. Systematic or multiple reproduction, distribution to multiple locations via electronic or other means, duplication of any material in this paper for a fee or for commercial purposes, or modification of the content of the paper is prohibited and is subject to penalties under law.