Constructive Approximation

, Volume 2, Issue 1, pp 331–337 | Cite as

Steckin-marchaud-type inequalities in connection with bernstein polynomials

  • Erich van Wickeren


The purpose of this paper is to derive the estimate (0≤α≤2,n∈N,ϕ(x)=[x(1−x)]1/2)
$$\omega _\alpha (n^{ - {1 \mathord{\left/ {\vphantom {1 2}} \right. \kern-\nulldelimiterspace} 2}} ,f) \leqslant M_\alpha n^{ - 1} \sum\limits_{k = 1}^n {\left\| {\varphi ^{ - \alpha } (B_k f - f)} \right\|} c$$
in terms of the modulus of continuity (of second order)
$$\omega _\alpha (t,f): = \sup \{ \varphi ^{ - \alpha } (x)|\Delta _{h\varphi (x)}^ * f(x)|:x,x \pm h\varphi (x) \in [0,1],0< h \leqslant t\} $$
and the Bernstein polynomial Bnf for ϕ−αf∈C[0,1].

AMS classification

41A27 41A17 41A36 

Key words and phrases

Bernstein polynomials Modulus of continuity Steckin-Marchaud-type inequalities 


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  1. 1.
    M. Becker (1979):An elementary proof of the inverse theorem for Bernstein polynomials. Aequationes Math.,19:145–150.Google Scholar
  2. 2.
    H. Berens, G. G. Lorentz (1972):Inverse theorems for Bernstein polynomials. Indiana Univ. Math. J.,21:693–708.Google Scholar
  3. 3.
    R. A. DeVore, S. D. Riemenschneider, R. C. Sharpley (1979):Weak interpolation in Banach spaces. S. Funct. Anal.,33:58–94.Google Scholar
  4. 4.
    Z. Ditzian (1979):A global inverse theorem for combinations of Bernstein polynomials. J. Approx. Theory,26:277–292.Google Scholar
  5. 5.
    Z. Ditzian (1980):On interpolation of Lp[a, b] and weighted Sobolev spaces. Pacific J. Math.,90:307–323.Google Scholar
  6. 6.
    K. G. Ivanov (1982):On Bernstein polynomials. C. R. Acad. Bulgare Sci.35:893–896.Google Scholar
  7. 7.
    G. G. Lorentz (1964):Inequalities and the saturation classes of Bernstein polynomials. In: On Approximation Theory. Proc. Conf. Oberwolfach 1963 (P. L. Butzer, J. Korevaar, eds.). Basel: Birkhäuser, pp. 200–207.Google Scholar
  8. 8.
    V. Totik (1984):An interpolation theorem and its application to positive operators. Pacific J. Math.,111:447–481.Google Scholar
  9. 9.
    V. Totik (1984):The necessity of a new kind of modulus of smoothness. In: Anniversary Volume on Approximation Theory and Functional Analysis. Proc. Conf. Oberwolfach 1983 (P. L. Butzer et al., eds.). Basel: Birkhäuser, pp. 233–248.Google Scholar

Copyright information

© Springer-Verlag New York Inc. 1986

Authors and Affiliations

  • Erich van Wickeren
    • 1
  1. 1.Lehrstuhl A für MathematikRhein-Westf. Techn. Hochschule AachenAachenF.R.G.

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