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Quantitative theorems on approximation by Bernstein-Stancu operators

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Abstract

In 1972 D. D. Stancu introduced a generalization\(L_{mp} ^{< \alpha \beta \gamma > }\) of the classical Bernstein operators given by the formula

$$L_{mp}< \alpha \beta \gamma > (f,x) = \sum\limits_{k = 0}^{m + p} {\left( {\begin{array}{*{20}c} {m + p} \\ k \\ \end{array} } \right)} \frac{{x^{(k, - \alpha )} \cdot (1 - x)^{(m + p - k, - \alpha )} }}{{1^{(m + p, - \alpha )} }}f\left( {\frac{{k + \beta }}{{m + \gamma }}} \right)$$

.

Special cases of these operators had been investigated before by quite a number of authors and have been under investigation since then.

The aim of the present paper is to prove general results for all positiveL <αβγ> mp 's as far as direct theorems involving different kinds of moduli of continuity are concerned.

When applied to special cases considered previously, all our corollaries of the general theorems will be as good as or yield improvements of the known results. All estimates involving the second order modulus of continuity are new.

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Authors and Affiliations

  1. Department of Mathematics, University of Duisburg, D-4100, Duisburg, 1, West Germany

    J. Meier

  2. Department of Mathematics and Computer Science, Drexel University, 19104, Philadelphia, PA, USA

    H. H. Gonska

Authors
  1. H. H. Gonska
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  2. J. Meier
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Gonska, H.H., Meier, J. Quantitative theorems on approximation by Bernstein-Stancu operators. Calcolo 21, 317–335 (1984). https://doi.org/10.1007/BF02576170

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