A representation of the interpolation polynomial


We provide a decomposition formula for the classical polynomial interpolation operator and obtain the generalized Hermite interpolant through a limiting process. As a consequence of our results, we obtain/reobtain known and new identities related to interpolation theory.

This is a preview of subscription content, access via your institution.


  1. 1.

    Ampère, A.M.: Essai sur un nouveau mode d’exposition des principes du calcul différentiel, du calcul aux différences et de l’interpolation des suites, considérées comme dérivant d’une source commune. Ann. Math. Pures Appl. (Gergonne) 16, 329–349 (1825)

    MathSciNet  Google Scholar 

  2. 2.

    Atiken, A.: Determinants and Matrices. Oliver and Boyd, Edinburgh (1956)

    Google Scholar 

  3. 3.

    Brezinski, C.: The Mühlbach-Neville-Aitken algorithm and some extensions. BIT 20(4), 444–451 (1980). https://doi.org/10.1007/BF01X00000.933638

    MathSciNet  Article  Google Scholar 

  4. 4.

    Brezinski, C.: Recursive interpolation, extrapolation and projection. J. Comput. Appl. Math. 9(4), 369–376 (1983). https://doi.org/10.1016/0377-0427(83)90008-0

    MathSciNet  Article  Google Scholar 

  5. 5.

    Brezinski, C.: Some determinantal identities in a vector space, with applications. In: Padé Approximation and its Applications, Bad Honnef 1983 (Bad Honnef, 1983), Lecture Notes in Math, vol. 1071, pp 1–11. Springer, Berlin (1984). https://doi.org/10.1007/BFb0099606

  6. 6.

    Brezinski, C.: Other manifestations of the Schur complement. Linear Algebra Appl. 111, 231–247 (1988). https://doi.org/10.1016/0024-3795(88)90062-6

    MathSciNet  Article  Google Scholar 

  7. 7.

    Brezinski, C.: History of continued fractions and Padé approximants. Springer Series in Computational Mathematics, vol. 12. Springer, Berlin (1991). https://doi.org/10.1007/978-3-642-58169-4

    Google Scholar 

  8. 8.

    Brezinski, C.: The generalizations of Newton’s interpolation formula due to Mühlbach and Andoyer. Electron. Trans. Numer. Anal. 2(Sept.), 130–137 (1994)

    MathSciNet  MATH  Google Scholar 

  9. 9.

    Brezinski, C., Redivo Zaglia, M.: Extrapolation Methods, Studies in Computational Mathematics, vol. 2. North-Holland Publishing Co., Amsterdam (1991). Theory and practice, With 1 IBM-PC floppy disk (5.25 inch)

    Google Scholar 

  10. 10.

    DeVore, R.A., Lorentz, G.G.: Constructive Approximation. Springer, Berlin (1993)

    Google Scholar 

  11. 11.

    Gasca, M., López-Carmona, A.: A general recurrence interpolation formula and its applications to multivariate interpolation. J. Approx. Theory 34 (4), 361–374 (1982). https://doi.org/10.1016/0021-9045(82)90078-8

    MathSciNet  Article  Google Scholar 

  12. 12.

    Hakopian, H.A.: Multivariate divided differences and multivariate interpolation of Lagrange and Hermite type. J. Approx. Theory 34(3), 286–305 (1982). https://doi.org/10.1016/0021-9045(82)90019-3

    MathSciNet  Article  Google Scholar 

  13. 13.

    Horwitz, A.: Means, generalized divided differences, and intersections of osculating hyperplanes. J. Math. Anal. Appl. 200(1), 126–148 (1996). https://doi.org/10.1006/jmaa.1996.0195

    MathSciNet  Article  Google Scholar 

  14. 14.

    Johansen, P.: Über osculierende interpolation. Scand. Actuar. J. 1931(4), 231–237 (1931). https://doi.org/10.1080/03461238.1931.10405852

    Article  Google Scholar 

  15. 15.

    Messaoudi, A., Sadaka, R., Sadok, H.: New algorithm for computing the Hermite interpolation polynomial. Numer. Algorithms 77(4), 1069–1092 (2018). https://doi.org/10.1007/s11075-017-0353-6

    MathSciNet  Article  Google Scholar 

  16. 16.

    Moritz, R.E.: On Taylor’s interpolation formula as a limiting case of the interpolation formula of Lagrange. J. Am. Stat. Assoc. 18(142), 781–784 (1923). http://www.jstor.org/stable/2276717

    Article  Google Scholar 

  17. 17.

    Mühlbach, G.: The general recurrence relation for divided differences and the general Newton-interpolation-algorithm with applications to trigonometric interpolation. Numer. Math. 32(4), 393–408 (1979). https://doi.org/10.1007/BF01401043

    MathSciNet  Article  Google Scholar 

  18. 18.

    Newton, I.: Philosophiae Naturalis Principia Mathematica. Printed by Joseph Streater by order of the Royal Society, London (1687)

  19. 19.

    Nörlund, N. E.: Leçons sur les séries d’interpolation. Gauthier-Villars et Cie, Paris (1926)

    Google Scholar 

  20. 20.

    Popoviciu, T.: Sur le reste dans certaines formules linéaires d’approximation de l’analyse. Mathematica (Cluj) 1(24), 95–142 (1959)

    MathSciNet  MATH  Google Scholar 

  21. 21.

    Salzer, H.E.: Hermite’s general osculatory interpolation formula and a finite difference analogue. J. Soc. Indust. Appl. Math. 8, 18–27 (1960)

    MathSciNet  Article  Google Scholar 

  22. 22.

    Simonsen, W.: On divided differences and osculatory interpolation. Skand. Aktuarietidskr. 31, 157–164 (1948). https://doi.org/10.1080/03461238.1948.10404897

    MathSciNet  MATH  Google Scholar 

  23. 23.

    Spitzbart, A.: A generalization of Hermite’s interpolation formula. Amer. Math. Monthly 67, 42–46 (1960). https://doi.org/10.2307/2308924

    MathSciNet  Article  Google Scholar 

Download references

Author information



Corresponding author

Correspondence to Mircea Ivan.

Additional information

Publisher’s note

Springer Nature remains neutral with regard to jurisdictional claims in published maps and institutional affiliations.

Rights and permissions

Reprints and Permissions

About this article

Verify currency and authenticity via CrossMark

Cite this article

Ivan, M., Neagos, V. A representation of the interpolation polynomial. Numer Algor (2021). https://doi.org/10.1007/s11075-021-01072-2

Download citation


  • Interpolation

Mathematics subject classification (2010)

  • 41A05