Optimal Points for Cubature Rules and Polynomial Interpolation on a Square



The nodes of certain minimal cubature rule are real common zeros of a set of orthogonal polynomials of degree n. They often consist of a well distributed set of points and interpolation polynomials based on them have desired convergence behavior. We report what is known and the theory behind by explaining the situation when the domain of integrals is a square.



The author thanks two anonymous referees for their careful reading and corrections. The work was supported in part by NSF Grant DMS-1510296.


  1. 1.
    Bojanov, B., Petrova, G.: On minimal cubature formulae for product weight function. J. Comput. Appl. Math. 85, 113–121 (1997)MathSciNetCrossRefGoogle Scholar
  2. 2.
    Bos, L., Caliari, M., De Marchi, S., Vianello, M., Xu, Y.: Bivariate Lagrange interpolation at the Padua points: the generating curve approach. J. Approx. Theory 143, 15–25 (2006)MathSciNetCrossRefGoogle Scholar
  3. 3.
    Bos, L., De Marchi, S., Vianello, M.: On the Lebesgue constant for the Xu interpolation formula. J. Approx. Theory 141, 134–141 (2006)MathSciNetCrossRefGoogle Scholar
  4. 4.
    Bos, L., De Marchi, S., Vianello, M., Xu, Y.: Bivariate Lagrange interpolation at the Padua points: the ideal theory approach. Numer. Math. 108, 43–57 (2007)MathSciNetCrossRefGoogle Scholar
  5. 5.
    Cools, R.: Monomial cubature rules since “Stroud”: a compilation – part 2. J. Comput. Appl. Math. 112, 21–27 (1999)MathSciNetCrossRefGoogle Scholar
  6. 6.
    Cools, R., Rabinowitz, P.: Monomial cubature rules since “Stroud”: a compilation. J. Comput. Appl. Math. 48, 309–326 (1993)MathSciNetCrossRefGoogle Scholar
  7. 7.
    Dick, J., Kuo, F., Sloan, I.: High-dimensional integration: the quasi-Monte Carlo way. Acta Numer. 22, 133–288 (2013)MathSciNetCrossRefGoogle Scholar
  8. 8.
    Dunkl, C.F., Xu, Y.: Orthogonal Polynomials of Several Variables. Encyclopedia of Mathematics and its Applications, vol. 155. Cambridge University Press, Cambridge (2014)Google Scholar
  9. 9.
    Erb, W.: Bivariate Lagrange interpolation at the node points of Lissajous curves – the degenerate case. Appl. Math. Comput. 289, 409–425 (2016)MathSciNetCrossRefGoogle Scholar
  10. 10.
    Erb, W., Kaethner, C., Ahlborg, M., Buzug, T.M.: Bivariate Lagrange interpolation at the node points of non–degenerate Lissajous curves. Numer. Math. 133, 685–705 (2016)MathSciNetCrossRefGoogle Scholar
  11. 11.
    Fischer, G.: Plane Algebraic Curves, translated by Leslie Kay. American Mathematical Society (AMS), Providence, RI (2001)Google Scholar
  12. 12.
    Harris, L.: Bivariate Lagrange interpolation at the Chebyshev nodes. Proc. Am. Math. Soc. 138, 4447–4453 (2010)MathSciNetCrossRefGoogle Scholar
  13. 13.
    Harris, L.: Bivariate polynomial interpolation at the Geronimus nodes. In: Complex analysis and dynamical systems V. Contemporary Mathematics, vol. 591, pp. 135–147. American Mathematical Society, Providence, RI (2013), Israel Mathematical Conference ProceedingsGoogle Scholar
  14. 14.
    Harris, L.: Lagrange polynomials, reproducing kernels and cubature in two dimensions. J. Approx. Theory, 195, 43–56 (2015)MathSciNetCrossRefGoogle Scholar
  15. 15.
    Koornwinder, T.H.: Orthogonal polynomials in two variables which are eigenfunctions of two algebraically independent partial differential operators, I, II. Proc. Kon. Akad. v. Wet., Amsterdam 36, 48–66 (1974)Google Scholar
  16. 16.
    Li, H., Sun, J., Xu, Y.: Cubature formula and interpolation on the cubic domain. Numer. Math. Theory Methods Appl. 2, 119–152 (2009)Google Scholar
  17. 17.
    Möller, H.: Kubaturformeln mit minimaler Knotenzahl. Numer. Math. 25, 185–200 (1976)Google Scholar
  18. 18.
    Morrow, C.R., Patterson, T.N.L.: Construction of algebraic cubature rules using polynomial ideal theory. SIAM J. Numer. Anal. 15, 953–976 (1978)MathSciNetCrossRefGoogle Scholar
  19. 19.
    Mysovskikh, I.P.: Numerical characteristics of orthogonal polynomials in two variables. Vestnik Leningrad Univ. Math. 3, 323–332 (1976)Google Scholar
  20. 20.
    Mysovskikh, I.P.: Interpolatory Cubature Formulas. Nauka, Moscow (1981)Google Scholar
  21. 21.
    Schmid, H.: On cubature formulae with a minimal number of knots. Numer. Math. 31, 282–297 (1978)MathSciNetCrossRefGoogle Scholar
  22. 22.
    Schmid, H., Xu, Y.: On bivariate Gaussian cubature formula. Proc. Am. Math. Soc. 122, 833–842 (1994)MathSciNetCrossRefGoogle Scholar
  23. 23.
    Stroud, A.H.: Approximate Calculation of Multiple Integrals. Prentice-Hall, Englewood Cliffs, N.J. (1971)Google Scholar
  24. 24.
    Szili, L., Vértesi, P.: On multivariate projection operators. J. Approx. Theory 159, 154–164 (2009)MathSciNetCrossRefGoogle Scholar
  25. 25.
    Xu, Y.: Gaussian cubature and bivariable polynomial interpolation. Math. Comput. 59, 547–555 (1992)MathSciNetCrossRefGoogle Scholar
  26. 26.
    Xu, Y.: Common Zeros of Polynomials in Several Variables and Higher Dimensional Quadrature. Pitman Research Notes in Mathematics Series, Longman, Essex (1994)Google Scholar
  27. 27.
    Xu, Y.: On zeros of multivariate quasi-orthogonal polynomials and Gaussian cubature formulae. SIAM J. Math. Anal. 25, 991–1001 (1994)MathSciNetCrossRefGoogle Scholar
  28. 28.
    Xu, Y.: Lagrange interpolation on Chebyshev points of two variables. J. Approx. Theory 87, 220–238 (1996)MathSciNetCrossRefGoogle Scholar
  29. 29.
    Xu, Y.: Minimal Cubature rules and polynomial interpolation in two variables. J. Approx. Theory 164, 6–30 (2012)MathSciNetCrossRefGoogle Scholar
  30. 30.
    Xu, Y.: Orthogonal polynomials and expansions for a family of weight functions in two variables. Constr. Approx. 36, 161–190 (2012)MathSciNetCrossRefGoogle Scholar
  31. 31.
    Xu, Y.: Minimal Cubature rules and polynomial interpolation in two variables, II. J. Approx. Theory 214, 49–68 (2017)MathSciNetCrossRefGoogle Scholar

Copyright information

© Springer International Publishing AG, part of Springer Nature 2018

Authors and Affiliations

  1. 1.Department of MathematicsUniversity of OregonEugeneUSA

Personalised recommendations