Bivariate Lagrange interpolation at the Chebyshev nodes
Author:
Lawrence A. Harris
Journal:
Proc. Amer. Math. Soc. 138 (2010), 44474453
MSC (2010):
Primary 65D05, 65D32; Secondary 33C50, 41A05, 42B05
Published electronically:
July 15, 2010
MathSciNet review:
2680069
Fulltext PDF
Abstract 
References 
Similar Articles 
Additional Information
Abstract: We discuss Lagrange interpolation on two sets of nodes in two dimensions where the coordinates of the nodes are Chebyshev points having either the same or opposite parity. We use a formula of Xu for Lagrange polynomials to obtain a general interpolation theorem for bivariate polynomials at either set of Chebyshev nodes. An extra term must be added to the interpolation formula to handle all polynomials with the same degree as the Lagrange polynomials. We express this term as a specifically determined linear combination of canonical polynomials that vanish on the set of Chebyshev nodes being considered. As an application we deduce in an elementary way known minimal and near minimal cubature formulas applying to both the even and the odd Chebyshev nodes. Finally, we restrict to triangular subsets of the Chebyshev nodes to show unisolvence and deduce a Lagrange interpolation formula for bivariate symmetric and skewsymmetric polynomials. This result leads to another proof of the interpolation formula.
 1.
Borislav
Bojanov and Guergana
Petrova, On minimal cubature formulae for product weight
functions, J. Comput. Appl. Math. 85 (1997),
no. 1, 113–121. MR 1482159
(98m:65032), 10.1016/S03770427(97)001337
 2.
Len
Bos, Marco
Caliari, Stefano
De Marchi, Marco
Vianello, and Yuan
Xu, Bivariate Lagrange interpolation at the Padua points: the
generating curve approach, J. Approx. Theory 143
(2006), no. 1, 15–25. MR 2271722
(2007h:41001), 10.1016/j.jat.2006.03.008
 3.
Len
Bos, Stefano
De Marchi, Marco
Vianello, and Yuan
Xu, Bivariate Lagrange interpolation at the Padua points: the ideal
theory approach, Numer. Math. 108 (2007), no. 1,
43–57. MR
2350184 (2008j:41001), 10.1007/s002110070112z
 4.
Marco
Caliari, Stefano
De Marchi, and Marco
Vianello, Hyperinterpolation on the square, J. Comput. Appl.
Math. 210 (2007), no. 12, 78–83. MR 2389158
(2009c:65018), 10.1016/j.cam.2006.10.058
 5.
Lawrence
A. Harris, Multivariate Markov polynomial inequalities and
Chebyshev nodes, J. Math. Anal. Appl. 338 (2008),
no. 1, 350–357. MR 2386420
(2008m:41003), 10.1016/j.jmaa.2007.05.044
 6.
, A proof of Markov's theorem for polynomials on Banach spaces, J. Math. Anal. Appl. 368 (2010), 374381.
 7.
Yuan
Xu, Common zeros of polynomials in several variables and
higherdimensional quadrature, Pitman Research Notes in Mathematics
Series, vol. 312, Longman Scientific & Technical, Harlow, 1994. MR 1413501
(97m:41002)
 8.
Yuan
Xu, Lagrange interpolation on Chebyshev points of two
variables, J. Approx. Theory 87 (1996), no. 2,
220–238. MR 1418495
(97k:41006), 10.1006/jath.1996.0102
 1.
 B. Bojanov and G. Petrova, On minimal cubature formulae for product weight functions, J. Comput. Appl. Math. 85 (1997), 113121. MR 1482159 (98m:65032)
 2.
 L. Bos, M. Caliari, S. De Marchi, M. Vianello, and Y. Xu, Bivariate Lagrange interpolation at the Padua points: the generating curve approach, J. Approx. Theory 143 (2006), 1525. MR 2271722 (2007h:41001)
 3.
 L. Bos, S. De Marchi, M. Vianello, and Y. Xu, Bivariate Lagrange interpolation at the Padua points: the ideal theory approach, Numer. Math. 108 (2007), 4357. MR 2350184 (2008j:41001)
 4.
 M. Caliari, S. De Marchi, and M. Vianello, Hyperinterpolation on the square, J. Comput. Appl. Math. 210 (2007), 7883. MR 2389158 (2009c:65018)
 5.
 L. A. Harris, Multivariate Markov polynomial inequalities and Chebyshev nodes, J. Math. Anal. Appl. 338 (2008), 350357. MR 2386420 (2008m:41003)
 6.
 , A proof of Markov's theorem for polynomials on Banach spaces, J. Math. Anal. Appl. 368 (2010), 374381.
 7.
 Yuan Xu, Common Zeros of Polynomials in Several Variables and HigherDimensional Quadrature, Pitman Research Notes in Mathematics, Longman, Essex, 1994. MR 1413501 (97m:41002)
 8.
 , Lagrange interpolation on Chebyshev points of two variables, J. Approx. Theory 87 (1996), 220238. MR 1418495 (97k:41006)
Similar Articles
Retrieve articles in Proceedings of the American Mathematical Society
with MSC (2010):
65D05,
65D32,
33C50,
41A05,
42B05
Retrieve articles in all journals
with MSC (2010):
65D05,
65D32,
33C50,
41A05,
42B05
Additional Information
Lawrence A. Harris
Affiliation:
Department of Mathematics, University of Kentucky, Lexington, Kentucky 40506
Email:
larry@ms.uky.edu
DOI:
http://dx.doi.org/10.1090/S000299392010105816
Received by editor(s):
April 5, 2009
Received by editor(s) in revised form:
March 6, 2010
Published electronically:
July 15, 2010
Communicated by:
Walter Van Assche
Article copyright:
© Copyright 2010
American Mathematical Society
The copyright for this article reverts to public domain 28 years after publication.
