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Constructive Approximation

, Volume 1, Issue 1, pp 137–154 | Cite as

Generalized monosplines and optimal approximation

  • Nira Dyn
Article

Abstract

Generalized monosplines of least norm are shown to exist and to determine optimal approximation processes such as numerical integration, interpolation and best approximating spaces. This extends various classical results related to monosplines and perfect splines, which are particular cases of generalized monosplines. The analysis here also provides for a unified treatment of the two classical classes of monosplines and perfect splines of least norm, and of their extremal properties.

AMS classification

41A15 41A5 41A65 

Key words and phrases

Perfect splines Monosplines Extended totally positive kernels Monotone norms Optimal quadrature formulas Optimal interpolation N-widths Optimal spaces 

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References

  1. 1.
    R. B. Barrar, H. L. Loeb (1981):Oscillating Tchebycheff systems. J. Approx. Theory,31:188–197.Google Scholar
  2. 2.
    S. N. Bernstein (1937): Extremal Properties of the Polynomial of Best Approximation of a Continuous Function. Leningrad: [In Russian].Google Scholar
  3. 3.
    B. D. Bojanov (1977):Existence of extended monosplines of least deviation. SERDICA Bulg. Math. Pub.,3:261–272.Google Scholar
  4. 4.
    B. D. Bojanov, D. Braess, N. Dyn (In press):Generalized Gaussian quadrature formulas. J. Approx. Theory.Google Scholar
  5. 5.
    D. Braess, N. Dyn (1982):On the uniqueness of monosplines and perfect splines of least L 1- andL 2-norms. J. Analyse Math.,41:217–233.Google Scholar
  6. 6.
    N. Dyn (Richter) (1979):On best nonlinear approximation in sign-monotone norms and in norms induced by inner products. SIAM J. Numer. Anal.,16:612–622.Google Scholar
  7. 7.
    N. Dyn (1983):Perfect splines of minimum norm for monotone norms and norms induced by inner products; with applications to tensor-product approximation and n-width of integral operators. J. Approx. Theory,38:105–138.Google Scholar
  8. 8.
    S. Karlin (1968): Total Positivity, vol. I. Stanford, California: Stanford University Press.Google Scholar
  9. 9.
    S. Karlin (1976):On a class of best nonlinear approximation problems and extended monosplines. In: Studies in Spline Functions and Approximation Theory. New York: Academic Press, pp. 19–66.Google Scholar
  10. 10.
    E. Kimchi, N. Richter-Dyn (1978):Restricted range approximation of k-convex functions in monotone norms. SIAM J. Numer. Anal.,15:1030–1038.Google Scholar
  11. 11.
    E. Kimchi, N. Richter-Dyn (1979):A necessary condition for best approximation in monotone and sign-monotone norms. J. Approx. Theory,25:169–175.Google Scholar
  12. 12.
    C. A. Micchelli, A. Pinkus (1978):Some problems in the approximation of functions of two variables and n-widths of integral operators. J. Approx. Theory,24:51–77.Google Scholar
  13. 13.
    C. A. Micchelli, T. J. Rivlin (1977):A survey of optimal recovery. In: Optimal Estimation in Approximation Theory (C. A. Micchelli, T. J. Rivlin, eds.). New York: Plenum Press, pp. 1–54.Google Scholar
  14. 14.
    A. Pinkus, (1981):Bernstein comparison theorem and a problem of Braess. Aequationes Math.,22:318–320.Google Scholar
  15. 15.
    I. J. Schoenberg (1969):Monosplines and quadrature formulae. In: Theory and Applications of Spline Functions (T. N. E. Greville, ed.). New York: Academic Press, pp. 157–207.Google Scholar
  16. 16.
    V.M. Tihomirov (1969):Best methods of approximation and interpolation of differentiable functions in the space C[−1,1]. Math. U.S.S.R. Sbornik,9:275–289.Google Scholar

Copyright information

© Springer-Verlag New York Inc 1985

Authors and Affiliations

  • Nira Dyn
    • 1
  1. 1.School of Mathematical SciencesTel-Aviv UniversityTel-AvivIsrael

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