Numerische Mathematik

, Volume 26, Issue 2, pp 191–200 | Cite as

The optimal recovery of smooth functions

  • C. A. Micchelli
  • T. J. Rivlin
  • S. Winograd


It is shown that there is a positive lower bound,c, to the uniform error in any scheme designed to recover all functions of a certain smoothness from their values at a fixed finite set of points. This lower bound is essentially attained by interpolation at the points by splines with canonical knots. Estimates ofc are also given.


Smooth Function Mathematical Method Optimal Recovery Uniform Error 
These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.


Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.


  1. 1.
    Beesack, P.R.: On the Green's function of an N-point boundary value problem. Pacific J. Math.,12, 801–812 (1962)Google Scholar
  2. 2.
    Cavaretta, A.S. Jr.: Oscillation and zero properties for perfect splines and monosplines. J. d'Analyse Math.28, 41–59 (1975)Google Scholar
  3. 3.
    Das, K.M., Vatsala, A.S.: On Green's function of ann-point boundary value problem. Trans. A.M.S.,182, 469–480 (1973)Google Scholar
  4. 4.
    de Boor, C.: A remark concerning perfect splines. Bull. A.M.S.,80, 724–727 (1974)Google Scholar
  5. 5.
    de La Vallée Poussin, C.J.: Leçons sur l'Approximation des Fonctions d'une Variable Réele. Paris: Gauthier-Villars 1952Google Scholar
  6. 6.
    Karlin, S.: Some variational problems on certain Sobolev spaces and perfect splines, Bull. A.M.S.,79, 124–128 (1973). Also, Trans. A.M.S.,106, 25–66 (1975)Google Scholar
  7. 7.
    Schoenberg, I.J.: On spline functions, Inequalities. Academic Press, 1967, p. 255–291Google Scholar
  8. 8.
    Schoenberg, I.J.: Cardinal Spline Interpolation. S.I.A.M. Philadelphia, 1973Google Scholar
  9. 9.
    Tihomirov, V.M.: Best methods of approximation and interpolation of differentiable functions in the spaceC [−1,1] Math. USSR Sbornik,9, 275–289 (1969)Google Scholar

Copyright information

© Springer-Verlag 1976

Authors and Affiliations

  • C. A. Micchelli
    • 1
  • T. J. Rivlin
    • 1
  • S. Winograd
    • 1
  1. 1.Thomas J. Watson Research CenterIBMYorktown HeightsUSA

Personalised recommendations