Remarks on the Jackson and Whitney Constants

  • Borislav Bojanov
Part of the Mathematics and Its Applications book series (MAIA, volume 430)


The paper is devoted to the constants in the Jackson theorem about approximation of continuous functions by polynomials on [a, b] and the Whitney type estimation of the interpolation error. The Jackson theorem is derived here on the basis of the Tchebycheff alternation theorem. This approach leads to an algorithm for computation of the exact Jackson constant. In the second part we give a new representation of the remainder in the Lagrange interpolation formula and then use it to get estimates of Whitney type for certain classical approximation schemes.

Key words and phrases

Best approximation Degree of approximation Jackson theorem Whitney constant 


Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.


  1. 1.
    N. I. Akhiezer, Lectures on Approximation Theory, Nauka, Moscow, 1965. (Russian)Google Scholar
  2. 2.
    B. Bojanov, H. Hakopian and A. Sahakian, Spline Functions and Multivariate Interpolations, Kluwer, Dordrecht, 1993.zbMATHGoogle Scholar
  3. 3.
    B. Bojanov, A Jackson type theorem for Tchebyeheff systems, Math. Balkanica (to appear).Google Scholar
  4. 4.
    Yu. V. Kryakin, On the theorem of H. Whitney in spaces L p, 1 ≤ p ≤ ∞, Math. Balkanica 4(3) (1990), 258–271.MathSciNetzbMATHGoogle Scholar
  5. 5.
    C. A. Micchelli, On a numerically efficient method for computing multivariate B-splines, Multivariate Approximation (W. Schempp and K. Zeller, eds.), Birkhäuser Verlag, Basel, 1979, pp. 211-248.Google Scholar
  6. 6.
    I. P. Natanson, Constructive Function Theory, Gizdat, Moscow — Leningrad, 1948. (Russian)Google Scholar
  7. 7.
    E. Passow Another proof of Jackson’s theorem, J. Approx. Theory 3 (1970), 146–148.MathSciNetzbMATHCrossRefGoogle Scholar
  8. 8.
    Bl. Sendov On the theorem and constants of H. Whitney, Constr. Approx. 3 (1987), 1–11.MathSciNetzbMATHCrossRefGoogle Scholar
  9. 9.
    S. Tashev, On the distribution of points of maximal deviation, Approximation and Function Spaces (Z. Ciesielski, ed.), North-Holland, New York, 1981, pp. 791–799.Google Scholar
  10. 10.
    H. Whitney On functions with bounded n-th differences, J. Math. Pures Appl. 36 (1957), 67–95.MathSciNetzbMATHGoogle Scholar

Copyright information

© Springer Science+Business Media New York 1998

Authors and Affiliations

  • Borislav Bojanov
    • 1
  1. 1.Department of MathematicsUniversity of SofiaSofiaBulgaria

Personalised recommendations