, Volume 45, Issue 3, pp 177–192 | Cite as

New bivariate polynomial expansion with boundary data on the simplex

  • Francesco Aldo Costabile
  • Francesco Dell’Accio
  • Luca Guzzardi


We introduce an extension to the two-dimensional simplex of the univariate two-point expansion formula for sufficiently smooth real functions introduced in [13]; it is a polynomial expansion with algebraic degree of exactness. This expansion is applied to obtain a new class of embedded boundary-type cubature formulae on the simplex.


Approximation by polynomials interpolation simplex embedded cubature formulae 

Mathematics Subject Classification (2000)

Primary 41A05 41A10 65D05 65D15 Secondary 65D32 


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  1. 1.
    Abramowitz, M., Stegun, I.A. (eds.): Handbook of mathematical functions with formulas, graphs, and mathematical tables. Washington, D.C.: National Bureau of Standards 1972zbMATHGoogle Scholar
  2. 2.
    Agarwal, R.P., Wong, P.J.Y.: Error inequalities in polynomial interpolation and their applications. Dordrecht: Kluwer 1993zbMATHGoogle Scholar
  3. 3.
    Barnhill, R.E., Birkhoff, G., Gordon, W.J.: Smooth interpolation in triangles. J. Approx. Theory 8, 114–128 (1973)zbMATHCrossRefMathSciNetGoogle Scholar
  4. 4.
    Barnhill, R.E., Gregory, J.A.: Polynomial interpolation to boundary data on triangles. Math. Comp. 29, 726–735 (1975)zbMATHCrossRefMathSciNetGoogle Scholar
  5. 5.
    Boas, Jr., R.P.: Representation of functions by Lidstone series. DukeMath. J. 10, 239–245(1943)zbMATHMathSciNetGoogle Scholar
  6. 6.
    Buckholtz, J.D., Shaw, J.K.: On functions expandable in Lidstone series. J. Math. Anal. Appl. 47, 626–632 (1974)zbMATHCrossRefMathSciNetGoogle Scholar
  7. 7.
    Caira, R., Dell’Accio, F.: Shepard-Bernoulli operators. Math. Comp. 76, 299–321 (2007)zbMATHCrossRefMathSciNetGoogle Scholar
  8. 8.
    Costabile, F.A., Dell’Accio, F.: Expansions over a simplex of real functions by means of Bernoulli polynomials. Numer. Algorithms 28, 63–86 (2001)zbMATHCrossRefMathSciNetGoogle Scholar
  9. 9.
    Costabile, F.A., Dell’Accio, F.: Lidstone approximation on the triangle. Appl. Numer. Math. 52, 339–361 (2005)zbMATHCrossRefMathSciNetGoogle Scholar
  10. 10.
    Costabile, F.A., Dell’Accio, F.: Polynomial approximation of CM function by means of boundary values and applications: a survey. J. Comput. App. Math. 210, 116–135 (2007)zbMATHCrossRefMathSciNetGoogle Scholar
  11. 11.
    Costabile, F.A., Dell’Accio, F.: New embedded boundary-type quadrature formulas for the simplex. Numer. Algorithms 45, 253–267 (2007)zbMATHCrossRefMathSciNetGoogle Scholar
  12. 12.
    Costabile, F.A., Dell’Accio, F., Guzzardi, L.: New bivariate polynomial expansion with only boundary data. (Pubblicazioni LAN Unical 15) Arcavacata di Rende: Laboratorio di Analisi Numerica, Università della Calabria 2006Google Scholar
  13. 13.
    Costabile, F.A., Dell’Accio, F., Luceri, R.: Explicit polynomial expansions of regular real functions by means of even order Bernoulli polynomials and boundary values. J. Comput. Appl. Math. 176, 77–90 (2005)zbMATHCrossRefMathSciNetGoogle Scholar
  14. 14.
    Kergin, P.: A natural interpolation of C K functions. J.Approx. Theory 29, 278–293 (1980)zbMATHCrossRefMathSciNetGoogle Scholar
  15. 15.
    He, T.-X.: Dimensionality reducing expansion of multivariate integration. Boston: Birkhäuser Boston 2001, pp. 225zbMATHGoogle Scholar
  16. 16.
    Widder, D.V.: Completely convex functions and Lidstone series. Trans. Amer. Math. Soc. 51, 387–398 (1942)CrossRefMathSciNetGoogle Scholar
  17. 17.

Copyright information

© Springer-Verlag Italia 2008

Authors and Affiliations

  • Francesco Aldo Costabile
    • 1
  • Francesco Dell’Accio
    • 1
  • Luca Guzzardi
    • 1
  1. 1.Dipartimento di MatematicaUniversità della CalabriaRende (Cs)Italy

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