Constructive Approximation

, Volume 36, Issue 2, pp 161–190 | Cite as

Orthogonal Polynomials and Expansions for a Family of Weight Functions in Two Variables



Orthogonal polynomials for a family of weight functions on [−1,1]2,
$$\mathcal{W}_{\alpha,\beta,\gamma}(x,y) = |x+y|^{2\alpha+1}|x-y|^{2\beta+1} \bigl(1-x^2\bigr)^{\gamma}\bigl(1-y^2\bigr)^{\gamma},$$
are studied and shown to be related to the Koornwinder polynomials defined on the region bounded by two lines and a parabola. In the case of γ=±1/2, an explicit basis of orthogonal polynomials is given in terms of Jacobi polynomials, and a closed formula for the reproducing kernel is obtained. The latter is used to study the convergence of orthogonal expansions for these weight functions.


Orthogonal polynomials Orthogonal expansions Jacobi polynomials Two variables Lebesgue constants 

Mathematics Subject Classification (2000)

33C50 42C10 


  1. 1.
    Badkov, V.: Convergence in the mean and the almost everywhere of Fourier series in polynomials orthogonal on an interval. Math. USSR Sb. 24, 223–256 (1974) CrossRefGoogle Scholar
  2. 2.
    Beerends, R.J., Opdam, E.M.: Certain hypergeometric series related to the root system BC. Trans. Am. Math. Soc. 339, 581–609 (1993) MathSciNetMATHGoogle Scholar
  3. 3.
    Dai, F., Xu, Y.: Cesàro means of orthogonal expansions in several variables. Constr. Approx. 29, 129–155 (2009) MathSciNetMATHCrossRefGoogle Scholar
  4. 4.
    Dunkl, C.F., Xu, Y.: Orthogonal Polynomials of Several Variables. Encyclopedia of Mathematics and Its Applications, vol. 81. Cambridge University Press, Cambridge (2001) MATHCrossRefGoogle Scholar
  5. 5.
    Forrester, P.J., Warnaar, S.O.: The importance of the Selberg integral. Bull. Am. Math. Soc. 45, 489–534 (2008) MathSciNetMATHCrossRefGoogle Scholar
  6. 6.
    Heckman, G.J., Opdam, E.M.: Root systems and hypergeometric functions I. Compos. Math. 64, 329–352 (1987) MathSciNetMATHGoogle Scholar
  7. 7.
    Koornwinder, T.H.: Orthogonal polynomials in two variables which are eigenfunctions of two algebraically independent partial differential operators, I, II. Proc. K. Ned. Akad. Wet. 36, 48–66 (1974) MathSciNetGoogle Scholar
  8. 8.
    Koornwinder, T.H.: Two-variable analogues of the classical orthogonal polynomials. In: Askey, R.A. (ed.) Theory and Applications of Special Functions, pp. 435–495. Academic Press, New York (1975) Google Scholar
  9. 9.
    Koornwinder, T.H., Sprinkhuizen-Kuyper, I.: Generalized power series expansions for a class of orthogonal polynomials in two variables. SIAM J. Math. Anal. 9, 457–483 (1978) MathSciNetMATHCrossRefGoogle Scholar
  10. 10.
    Lorentz, G.: Approximation of Functions. Chelsea, New York (1986) MATHGoogle Scholar
  11. 11.
    Nevai, P.: Orthogonal polynomials. Mem. Am. Math. Soc. 18, 213 (1979) MathSciNetGoogle Scholar
  12. 12.
    Nevai, P.: Mean convergence of Lagrange interpolation III. Trans. Am. Math. Soc. 282, 669–698 (1984) MathSciNetMATHCrossRefGoogle Scholar
  13. 13.
    Schmid, H.J., Xu, Y.: On bivariate Gaussian cubature formula. Proc. Am. Math. Soc. 122, 833–842 (1994) MathSciNetMATHCrossRefGoogle Scholar
  14. 14.
    Sprinkhuizen-Kuyper, I.: Orthogonal polynomials in two variables. A further analysis of the polynomials orthogonal over a region bounded by two lines and a parabola. SIAM J. Math. Anal. 7, 501–518 (1976) MathSciNetMATHCrossRefGoogle Scholar
  15. 15.
    Szegő, G.: Orthogonal Polynomials, 4th edn. Am. Math. Soc. Colloq. Publ., vol. 23. Am. Math. Soc., Providence (1975) Google Scholar
  16. 16.
    Vretare, L.: Formulas for elementary spherical functions and generalized Jacobi polynomials. SIAM J. Math. Anal. 15, 805–833 (1984) MathSciNetMATHCrossRefGoogle Scholar
  17. 17.
    Xu, Y.: Mean convergence of generalized Jacobi series and interpolating polynomials, I. J. Approx. Theory 72, 237–251 (1993) MathSciNetMATHCrossRefGoogle Scholar
  18. 18.
    Xu, Y.: Christoffel functions and Fourier Series for multivariate orthogonal polynomials. J. Approx. Theory 82, 205–239 (1995) MathSciNetMATHCrossRefGoogle Scholar
  19. 19.
    Xu, Y.: Lagrange interpolation on Chebyshev points of two variables. J. Approx. Theory 87, 220–238 (1996) MathSciNetMATHCrossRefGoogle Scholar

Copyright information

© Springer Science+Business Media, LLC 2012

Authors and Affiliations

  1. 1.Department of MathematicsUniversity of OregonEugeneUSA

Personalised recommendations