Skip to main content
SpringerLink
Log in

On some aspects of multivariate polynomial interpolation

  • Published:
Advances in Computational Mathematics Aims and scope Submit manuscript

Abstract

The purpose of this paper is to present some aspects of multivariate Hermite polynomial interpolation. We do not focus on algebraic considerations, combinatoric and geometric aspects, but on explicitation of formulas for uniform and non-uniform bivariate interpolation and some higher dimensional problems. The concepts of similar and equivalent interpolation schemes are introduced and some differential aspects related to them are also investigated.

This is a preview of subscription content, log in via an institution to check access.

Access this article

Log in via an institution

Price excludes VAT (USA)
Tax calculation will be finalised during checkout.

Instant access to the full article PDF.

Institutional subscriptions

Similar content being viewed by others

References

  1. T. Bloom, On the convergence of multivariate Lagrange interpolants, Constr. Approx. 5 (1989) 415–435.

    Article  MATH  MathSciNet  Google Scholar 

  2. J.R. Busch, Osculatory interpolation in Rn, SIAM J. Num. Anal. 22 (1985) 107–113.

    Article  MATH  MathSciNet  Google Scholar 

  3. C.K. Chung and T.H. Yao, On lattices admitting unique Lagrange interpolation, SIAM J. Numer. Anal. 14 (1977) 735–741.

    Article  MATH  MathSciNet  Google Scholar 

  4. C. Coatmelec, Approximation et Interpolation des fonctions différentiables de plusieurs variables, Ann. Sci. École Norm. Sup. 83 (1966) 271–341.

    MATH  MathSciNet  Google Scholar 

  5. P.J. Davis, Interpolation and Approximation (Blaisdell, New York, 1963).

    MATH  Google Scholar 

  6. C. de Boor and A. Ron, On multivariate polynomial Interpolation, Constr. Approx. 6 (1990).

  7. J. Dieudonné, Fondements de l'Analyse Moderne (Gauthiers-Villars, Paris, 1967).

    Google Scholar 

  8. M. Gasca and A. López-Carmona, A general recurrence formula and its applications to multivariate interpolation, J. Approx. Theory 34 (1982) 361–374.

    Article  MATH  MathSciNet  Google Scholar 

  9. M. Gasca and J.I. Maetzu, On Lagrange and Hermite interpolation in Rn, Numer. Math. 39 (1982) 1–14.

    Article  MATH  MathSciNet  Google Scholar 

  10. M. Gasca and T. Sauer, Polynomial interpolation in several variables, this issue.

  11. H.V. Gevorgian, H.A. Hakopian and A.A. Sahakian, On the bivariate Hermite interpolation problem, Preprint, Arbeitspapiere der GMD (1992).

  12. G. Glaeser, L'interpolation des fonctions différentiables, in: Proc. Liverpool Singularities Symposium II, Lecture Notes in Mathematics, Vol. 209 (Springer, Berlin, 1971) pp. 1–33.

    Article  MathSciNet  Google Scholar 

  13. R.B. Guenther and E.L. Roetman, Some observations on interpolation in higher dimensions, Math. Comp. 24 (1970).

  14. A. Le Méhauté, Taylor interpolation of order n at the vertices of a triangle. Application for Hermite interpolation and finite elements, in: Approximation Theory and Applications, ed. S. Ziegler (Academic Press, New York, 1981).

    Google Scholar 

  15. A. Le Méhauté, On Hermite elements of class C q in R3, Approximation Theory IV, eds. C.K. Chui, L.L. Schumaker and J. Ward (Academic Press, New York, 1983).

    Google Scholar 

  16. A. Le Méhauté, Interpolation et approximation par des fonctions polynomiales par morceaux dans Rn, Thèse de Doctorat es-Sciences, Université de Rennes (1984).

    Google Scholar 

  17. A. Le Méhauté, Unisolvent interpolation in Rn and the simplicial polynomial finite element method, in: Topics in Multivariate Approximation, eds. C.K. Chui, L.L. Schumaker and F.I. Utreras (Academic Press, 1987).

  18. A. Le Méhauté, An efficient algorithm for Ck simplicial finite element interpolation, in: Multivariate Approximation and Interpolation, eds. W. Haussman and K. Jetter (Birkhaüser, Basel, 1990) pp. 179–192.

    Google Scholar 

  19. A. Le Méhauté, On multivariate Hermite polynomial interpolation, Multivariate Approximations, from CAGD to Wavelets, eds. K. Jetter and F. Utreras (1993) pp. 179–192.

  20. S.L. Lee and G.M. Phillips, Interpolation on the triangle and simplex, in: Approximation Theory, Wavelets and Applications, ed. S.P. Singh (Kluwer Academic, 1995) pp. 177–196.

  21. G.G. Lorentz and R.A. Lorentz, Solvability problems of bivariate interpolation, Constr. Approx. 2 (1986).

  22. R.A. Lorentz, Uniform bivariate Hermite interpolation, in: Mathematical Methods in CAGD, eds. T. Lyche and L.L. Schumaker (Academic Press, Boston, 1989).

    Google Scholar 

  23. R.A. Lorentz, Multivariate Birkhoff Interpolation, Lecture Notes in Mathematics, Vol. 1516 (Springer, Berlin, 1992).

  24. S.H. Paskov, Singularity of bivariate interpolation, J. Approx. Theory 70 (1992).

  25. Th. Sauer and Y. Su, On multivariate Lagrange interpolation, Math. Comp. 64 (1995) 1147–1170.

    Article  MATH  MathSciNet  Google Scholar 

  26. I.J. Schoenberg, On Hermite–Birkhoff interpolation, J. Math. Anal. Appl. 16 (1966).

  27. H.C. Thacher, Jr. and W.E. Milne, Interpolation in several variables, J. SIAM 8 (1960) 33–42.

    MATH  MathSciNet  Google Scholar 

  28. R.S. Walker, Algebraic Curves (Springer, New York, 1978).

    MATH  Google Scholar 

Download references

Author information

Authors and Affiliations

  1. Département de Mathématiques, Université de Nantes, Faculté des Sciences et des Techniques, 2 rue de la Houssinière, B.P. 92208, F-44322, Nantes Cedex 03, France E-mail:

    A. Le Méhauté

Authors
  1. A. Le Méhauté
    View author publications

    You can also search for this author in PubMed Google Scholar

Rights and permissions

Reprints and permissions

About this article

Cite this article

Le Méhauté, A. On some aspects of multivariate polynomial interpolation. Advances in Computational Mathematics 12, 311–333 (2000). https://doi.org/10.1023/A:1018985606661

Download citation

  • Issue Date:

  • DOI: https://doi.org/10.1023/A:1018985606661

Search

Navigation