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Mathematics of Computation
Mathematics of Computation
ISSN 1088-6842(online) ISSN 0025-5718(print)

 

On the convergence rates of Legendre approximation


Authors: Haiyong Wang and Shuhuang Xiang
Journal: Math. Comp. 81 (2012), 861-877
MSC (2010): Primary 65D05, 65D99, 41A25
Published electronically: October 18, 2011
MathSciNet review: 2869040
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Abstract: The problem of the rate of convergence of Legendre approximation is considered. We first establish the decay rates of the coefficients in the Legendre series expansion and then derive error bounds of the truncated Legendre series in the uniform norm. In addition, we consider Legendre approximation with interpolation. In particular, we are interested in the barycentric Lagrange formula at the Gauss-Legendre points. Explicit barycentric weights, in terms of Gauss-Legendre points and corresponding quadrature weights, are presented that allow a fast evaluation of the Legendre interpolation formula. Error estimates for Legendre interpolation polynomials are also given.


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Additional Information

Haiyong Wang
Affiliation: Department of Applied Mathematics and Software, Central South University, Changsha, Hunan 410083, People’s Republic of China
Address at time of publication: Department of Computer Science, Katholieke Universiteit Leuven, Celestijnenlaan 200A, B-3001 Leuven, Belgium
Email: haiyong.wang@cs.kuleuven.be

Shuhuang Xiang
Affiliation: Department of Applied Mathematics and Software, Central South University, Changsha, Hunan 410083, People’s Republic of China
Email: xiangsh@mail.csu.edu.cn

DOI: http://dx.doi.org/10.1090/S0025-5718-2011-02549-4
PII: S 0025-5718(2011)02549-4
Keywords: Legendre expansion, Chebyshev expansion, Bernstein ellipse, barycentric Lagrange interpolation.
Received by editor(s): May 7, 2010
Received by editor(s) in revised form: February 17, 2011
Published electronically: October 18, 2011
Additional Notes: This work was supported by the NSF of China (No. 11071260).
Article copyright: © Copyright 2011 American Mathematical Society
The copyright for this article reverts to public domain 28 years after publication.