Bivariate Lagrange interpolation at the checkerboard nodes
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- by Lihua Cao, Srijana Ghimire and Xiang-Sheng Wang PDF
- Proc. Amer. Math. Soc. 150 (2022), 2153-2163 Request permission
Abstract:
In this paper, we derive an explicit formula for the bivariate Lagrange basis polynomials of a general set of checkerboard nodes. This formula generalizes existing results of bivariate Lagrange basis polynomials at the Padua nodes, Chebyshev nodes, Morrow-Patterson nodes, and Geronimus nodes. We also construct a subspace spanned by linearly independent bivariate vanishing polynomials that vanish at the checkerboard nodes and prove the uniqueness of the set of bivariate Lagrange basis polynomials in the quotient space defined as the space of bivariate polynomials with a certain degree over the subspace of bivariate vanishing polynomials.References
- 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, DOI 10.1016/S0377-0427(97)00133-7
- 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, DOI 10.1016/j.jat.2006.03.008
- 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, DOI 10.1007/s00211-007-0112-z
- Lawrence A. Harris, Bivariate Lagrange interpolation at the Chebyshev nodes, Proc. Amer. Math. Soc. 138 (2010), no. 12, 4447–4453. MR 2680069, DOI 10.1090/S0002-9939-2010-10581-6
- Lawrence A. Harris, Lagrange polynomials, reproducing kernels and cubature in two dimensions, J. Approx. Theory 195 (2015), 43–56. MR 3339053, DOI 10.1016/j.jat.2014.10.017
- Lawrence A. Harris, Alternation points and bivariate Lagrange interpolation, J. Comput. Appl. Math. 340 (2018), 43–52. MR 3807788, DOI 10.1016/j.cam.2018.02.014
- Lawrence A. Harris and Brian Simanek, Interpolation and cubature for rectangular sets of nodes, Proc. Amer. Math. Soc. 149 (2021), no. 8, 3485–3497. MR 4273151, DOI 10.1090/proc/15414
- C. R. Morrow and T. N. L. Patterson, Construction of algebraic cubature rules using polynomial ideal theory, SIAM J. Numer. Anal. 15 (1978), no. 5, 953–976. MR 507557, DOI 10.1137/0715062
- Yuan Xu, Lagrange interpolation on Chebyshev points of two variables, J. Approx. Theory 87 (1996), no. 2, 220–238. MR 1418495, DOI 10.1006/jath.1996.0102
Additional Information
- Lihua Cao
- Affiliation: College of Mathematics and Statistics, Shenzhen University, Shenzhen, Guangdong 518060, People’s Republic of China
- Srijana Ghimire
- Affiliation: Department of Mathematics, University of Louisiana at Lafayette, Lafayette, Louisiana 70503
- MR Author ID: 1441575
- ORCID: 0000-0003-0761-7639
- Xiang-Sheng Wang
- Affiliation: Department of Mathematics, University of Louisiana at Lafayette, Lafayette, Louisiana 70503
- MR Author ID: 824988
- ORCID: 0000-0003-0410-4643
- Email: xswang@louisiana.edu
- Received by editor(s): July 10, 2021
- Received by editor(s) in revised form: August 16, 2021, and August 20, 2021
- Published electronically: February 15, 2022
- Additional Notes: The first author was partially supported by National Natural Science Foundation of China (No. 11571375), the Natural Science Funding of Shenzhen University (No. 2018073), and the Shenzhen Scientific Research and Development Funding Program (No. JCYJ20170302144002028).
The third author is the corresponding author. - Communicated by: Mourad Ismail
- © Copyright 2022 American Mathematical Society
- Journal: Proc. Amer. Math. Soc. 150 (2022), 2153-2163
- MSC (2020): Primary 65D05
- DOI: https://doi.org/10.1090/proc/15834
- MathSciNet review: 4392350