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, Volume 29, Issue 2–4, pp 229–248 | Cite as

Linearkombinationen von iterierten Bernsteinoperatoren

  • Günter Felbecker


The Bernstein polynomials Bn(f) approximate every function f which is continuous on [0, 1] uniformly on [0, 1]. Also the derivatives of the Bernstein polynomials approach the derivatives of the function f uniformly on [0, 1], if f has continuous derivatives. In this paper we shall introduce polynomial operators, namely linear combinations of iterates of Bernstein operators, which have the same properties but, under definite conditions, approximate f more closely than the Bernstein operators.


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  1. 1.
    ABRAMOWITZ, M., and I. A. STEGUN: Handbook of mathematical functions. New York: Dover Publ. 1965Google Scholar
  2. 2.
    BUTZER, P. L.: Linearcombinations of Bernstein polynomials. Canad. J. Math.5, 559–567 (1953)Google Scholar
  3. 3.
    BUTZER, P. L., and R. J. NESSEL: Fourier Analysis and Approximation, Vol I. Basel: Birkhäuser 1971Google Scholar
  4. 4.
    DAVIS, P. J.: Interpolation and Approximation. New York: Blaisdell Publ. 1963Google Scholar
  5. 5.
    FRENTIU, M.: Linear combinations of Bernstein polynomials and of Mirakjan operators (Rumänisch). Studia Univ. Babes-Bolyai Ser. Math.-Mech.15, 63–68 (1970)Google Scholar
  6. 6.
    KARLIN, S., and Z. ZIEGLER: Iteration of positive approximation operators. J. Approximation Theory3, 310–339 (1970)Google Scholar
  7. 7.
    LORENTZ, G. G.: Bernstein polynomials. Toronto: University of Toronto Press 1953Google Scholar
  8. 8.
    LORENTZ, G. G.: Approximation of functions. New York-Chicago-San Francisco-Toronto-London: Holt, Rinehart and Winston 1966Google Scholar
  9. 9.
    MAY, C. P.: Saturation and inverse theorems for combinations of a class of exponential-type oprtators. Canad. J. Math.28, 1224–1250 (1976)Google Scholar
  10. 10.
    NATANSON, I. P.: On the approximation of multiply differentiable periodic functions by means of singular integrals (Russ.). Dokl. Akad. Nauk SSSR82, 337–339 (1952)Google Scholar

Copyright information

© Springer-Verlag 1979

Authors and Affiliations

  • Günter Felbecker
    • 1
  1. 1.Mathematisches Institut der RuhrUniversität BochumBochumBundesrepublik Deutschland

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