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Explicit barycentric weights for polynomial interpolation in the roots or extrema of classical orthogonal polynomials


Authors: Haiyong Wang, Daan Huybrechs and Stefan Vandewalle
Journal: Math. Comp. 83 (2014), 2893-2914
MSC (2010): Primary 41A05, 65D05, 65D15
DOI: https://doi.org/10.1090/S0025-5718-2014-02821-4
Published electronically: March 28, 2014
MathSciNet review: 3246814
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Abstract: Barycentric interpolation is arguably the method of choice for numerical polynomial interpolation. The polynomial interpolant is expressed in terms of function values using the so-called barycentric weights, which depend on the interpolation points. Few explicit formulae for these barycentric weights are known. In [H. Wang and S. Xiang, Math. Comp., 81 (2012), 861-877], the authors have shown that the barycentric weights of the roots of Legendre polynomials can be expressed explicitly in terms of the weights of the corresponding Gaussian quadrature rule. This idea was subsequently implemented in the Chebfun package [L. N. Trefethen and others, The Chebfun Development Team, 2011] and in the process generalized by the Chebfun authors to the roots of Jacobi, Laguerre and Hermite polynomials. In this paper, we explore the generality of the link between barycentric weights and Gaussian quadrature and show that such relationships are related to the existence of lowering operators for orthogonal polynomials. We supply an exhaustive list of cases, in which all known formulae are recovered and also some new formulae are derived, including the barycentric weights for Gauss-Radau and Gauss-Lobatto points. Based on a fast $ {\mathcal O}(n)$ algorithm for the computation of Gaussian quadrature, due to Hale and Townsend, this leads to an $ {\mathcal O}(n)$ computational scheme for barycentric weights.


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Additional Information

Haiyong Wang
Affiliation: School of Mathematics and Statistics, Huazhong University of Science and Technology, Wuhan 430074, People’s Republic of China
Email: haiyongwang@hust.edu.cn

Daan Huybrechs
Affiliation: Department of Computer Science, K.U.Leuven, Celestijnenlaan 200A, B-3001 Leuven, Belgium
Email: daan.huybrechs@cs.kuleuven.be

Stefan Vandewalle
Affiliation: Department of Computer Science, K.U.Leuven, Celestijnenlaan 200A, B-3001 Leuven, Belgium
Email: stefan.vandewalle@cs.kuleuven.be

DOI: https://doi.org/10.1090/S0025-5718-2014-02821-4
Received by editor(s): November 12, 2012
Received by editor(s) in revised form: March 22, 2013
Published electronically: March 28, 2014
Article copyright: © Copyright 2014 American Mathematical Society
The copyright for this article reverts to public domain 28 years after publication.