Constructive Approximation

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

Steckin-marchaud-type inequalities in connection with bernstein polynomials

  • Erich van Wickeren
Article

Abstract

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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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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