Interpolation and cubature for rectangular sets of nodes

Harris, Lawrence; Simanek, Brian

doi:10.1090/proc/15414

Interpolation and cubature for rectangular sets of nodes
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by Lawrence A. Harris and Brian Simanek PDF

Proc. Amer. Math. Soc. 149 (2021), 3485-3497 Request permission

Abstract:

This article obtains specific formulas for Lagrange polynomials for rectangular versions of the even and the odd product nodes in $\mathbb {R}^2$. These polynomials are applied to obtain an exact cubature formula for bivariate polynomials when the number of rows of the nodes exceeds the number of columns by one.

References

J. Baik, T. Kriecherbauer, K. T.-R. McLaughlin, and P. D. Miller, Discrete orthogonal polynomials, Annals of Mathematics Studies, vol. 164, Princeton University Press, Princeton, NJ, 2007. Asymptotics and applications. MR 2283089
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, 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
Stephen Deterding and Lawrence A. Harris, Interpolation and cubature at the Morrow-Patterson nodes generated by different Geronimus polynomials, Math. Proc. R. Ir. Acad. 117A (2017), no. 1, 5–12. MR 3662258, DOI 10.3318/pria.2017.117.02
Wolfgang Erb, Bivariate Lagrange interpolation at the node points of Lissajous curves—the degenerate case, Appl. Math. Comput. 289 (2016), 409–425. MR 3515864, DOI 10.1016/j.amc.2016.05.019
Mariano Gasca and Thomas Sauer, Polynomial interpolation in several variables, Adv. Comput. Math. 12 (2000), no. 4, 377–410. Multivariate polynomial interpolation. MR 1768957, DOI 10.1023/A:1018981505752
Mariano Gasca and Thomas Sauer, On the history of multivariate polynomial interpolation, J. Comput. Appl. Math. 122 (2000), no. 1-2, 23–35. Numerical analysis 2000, Vol. II: Interpolation and extrapolation. MR 1794650, DOI 10.1016/S0377-0427(00)00353-8
Lawrence A. Harris, Multivariate Markov polynomial inequalities and Chebyshev nodes, J. Math. Anal. Appl. 338 (2008), no. 1, 350–357. MR 2386420, DOI 10.1016/j.jmaa.2007.05.044
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
J. C. Mason and D. C. Handscomb, Chebyshev polynomials, Chapman & Hall/CRC, Boca Raton, FL, 2003. MR 1937591
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
Yuan Xu, Common zeros of polynomials in several variables and higher-dimensional quadrature, Pitman Research Notes in Mathematics Series, vol. 312, Longman Scientific & Technical, Harlow, 1994. MR 1413501