Numerical Algorithms

, Volume 61, Issue 2, pp 291–313 | Cite as

Radial orthogonality and Lebesgue constants on the disk

  • Annie Cuyt
  • Irem Yaman
  • Bayram Ali Ibrahimoglu
  • Brahim Benouahmane
Original   Paper


In polynomial interpolation, the choice of the polynomial basis and the location of the interpolation points play an important role numerically, even more so in the multivariate case. We explore the concept of spherical orthogonality for multivariate polynomials in more detail on the disk. We focus on two items: on the one hand the construction of a fully orthogonal cartesian basis for the space of multivariate polynomials starting from this sequence of spherical orthogonal polynomials, and on the other hand the connection between these orthogonal polynomials and the Lebesgue constant in multivariate polynomial interpolation on the disk. We point out the many links of the two topics under discussion with the existing literature. The new results are illustrated with an example of polynomial interpolation and approximation on the unit disk. The numerical example is also compared with the popular radial basis function interpolation.


Polynomial interpolation Lebesgue constant Orthogonal polynomials Unit disk Several variables 


Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.


  1. 1.
    Benouahmane, B., Cuyt, A.: Multivariate orthogonal polynomials, homogeneous Padé approximants and Gaussian cubature. Numer. Algorithms 24, 1–15 (2000). doi: 10.1023/A:1019128823463 MathSciNetzbMATHCrossRefGoogle Scholar
  2. 2.
    Benouahmane, B., Cuyt, A.: Properties of multivariate homogeneous orthogonal polynomials. J. Approx. Theory 113(1), 1–20 (2001). doi: 10.1006/jath.2000.3565 MathSciNetzbMATHCrossRefGoogle Scholar
  3. 3.
    Bloom, T., Bos, L.P., Calvi, J.P., Levenberg, N.: Polynomial interpolation and approximation in ℂd (2011).
  4. 4.
    Bojanov, B., Xu, Y.: On polynomial interpolation of two variables. J. Approx. Theory 120(3), 267–282 (2003). doi: 10.1016/S0021-9045(02)00023-0 MathSciNetzbMATHCrossRefGoogle Scholar
  5. 5.
    Bos, L., De Marchi, S., Caliari, M., Vianello, M., Xu, Y.: Bivariate Lagrange interpolation at the padua points: the generating curve approach. J. Approx. Theory 143(1), 15–25 (2006). doi: 10.1016/j.jat.2006.03.008 MathSciNetzbMATHCrossRefGoogle Scholar
  6. 6.
    Bos, L., De Marchi, S., Caliari, M., Vianello, M.: On the Lebesgue constant for the Xu interpolation formula. J. Approx. Theory 141(2), 134–141 (2006). doi: 10.1016/j.jat.2006.01.005 MathSciNetzbMATHCrossRefGoogle Scholar
  7. 7.
    Caliari, M., De Marchi, S., Vianello, M.: Bivariate polynomial interpolation on the square at new nodal sets. Appl. Math. Comput. 165, 261–274 (2005). doi: 10.1016/j.amc.2004.07.001 MathSciNetzbMATHCrossRefGoogle Scholar
  8. 8.
    Cuyt, A., Benouahmane, B., Hamsapriye, Yaman, I.: Symbolic-numeric Gaussian cubature rules. Appl. Numer. Math. 61(8), 929–945 (2011). doi: 10.1016/j.apnum.2011.03.003 MathSciNetzbMATHCrossRefGoogle Scholar
  9. 9.
    Gautschi, W.: On inverses of vandermonde and confluent vandermonde matrices. Numer. Math. 4(1), 117–123 (1962). doi: 10.1007/BF01386302 MathSciNetzbMATHCrossRefGoogle Scholar
  10. 10.
    Heinrichs, W.: Improved Lebesgue constants on the triangle. J. Comput. Phys. 207(2), 625–638 (2005). doi: 10.1016/ MathSciNetzbMATHCrossRefGoogle Scholar
  11. 11.
    Hesthaven, J.S.: From electrostatics to almost optimal nodal sets for polynomial interpolation in a simplex. SIAM J. Numer. Anal. 35(2), 655–676 (1998). doi: 10.1137/S003614299630587X MathSciNetzbMATHCrossRefGoogle Scholar
  12. 12.
    Humberto, R.: On the selection of the most adequate radial basis function. Appl. Numer. Math. 33(3), 1573–1583 (2009). doi: 10.1016/j.apm.2008.02.008 Google Scholar
  13. 13.
    Sauer, T., Xu, Y.: Regular points for lagrange interpolation on the unit disk. Numer. Algorithms 12(2), 287–296 (1996). doi: 10.1007/BF02142808 MathSciNetzbMATHCrossRefGoogle Scholar
  14. 14.
    Sündermann, B.: On projection constants of polynomial space on the unit ball in several variables. Math. Z. 188(1), 111–117 (1984). doi: 10.1007/BF0116387 MathSciNetzbMATHCrossRefGoogle Scholar
  15. 15.
    Szabados, J., Vértesi, P.: Interpolation of Functions. World Scientific, Teaneck (1990)zbMATHCrossRefGoogle Scholar
  16. 16.
    Xu, Y.: Funk–Hecke formulae for orthogonal polynomials on sphere and on balls. Bull. Lond. Math. Soc. 32(4), 447–457 (2000). doi: 10.1112/S0024609300007001 zbMATHCrossRefGoogle Scholar
  17. 17.
    Xu, Y.: Polynomial interpolation on the unit sphere and on the unit ball. Adv. Comput. Math. 20(1–3), 247–260 (2004). doi: 10.1023/A:1025851005416 MathSciNetzbMATHCrossRefGoogle Scholar

Copyright information

© Springer Science+Business Media, LLC 2012

Authors and Affiliations

  • Annie Cuyt
    • 1
  • Irem Yaman
    • 2
  • Bayram Ali Ibrahimoglu
    • 1
    • 3
  • Brahim Benouahmane
    • 4
  1. 1.Department of Mathematics and Computer ScienceUniversiteit Antwerpen (CMI)AntwerpenBelgium
  2. 2.Department of Mathematics, Faculty of ScienceGebze Institute of TechnologyGebzeTurkey
  3. 3.Department of Mathematical EngineeringYıldız Technical UniversityIstanbulTurkey
  4. 4.Département de Mathématiques, Faculté des Sciences et TechniquesUniversité Hassan IIMohammadiaMorocco

Personalised recommendations