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Direct and Converse Voronovskaya Estimates for the Bernstein Operator

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Abstract

As is known, if \(f\in C^2[0,1]\), then, for the Bernstein operator \(B_n\), there holds

$$\begin{aligned} \lim _{n\rightarrow \infty } n(B_n f(x)-f(x))=\frac{x(1-x)}{2}\,f''(x) \end{aligned}$$

uniformly on [0, 1]. We characterize the rate of this convergence in terms of K-functionals and moduli of smoothness.

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Authors and Affiliations

  1. Department of Mathematics and Informatics, University of Sofia, 5 James Bourchier Blvd., 1164, Sofia, Bulgaria

    Borislav R. Draganov & Ivan Gadjev

  2. Institute of Mathematics and Informatics, Bulgarian Academy of Sciences, bl. 8 Acad. G. Bonchev Str., 1113, Sofia, Bulgaria

    Borislav R. Draganov

Authors
  1. Borislav R. Draganov
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  2. Ivan Gadjev
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Correspondence to Borislav R. Draganov.

Additional information

This work was supported by Grant DN 02/14 of the Fund for Scientific Research of the Bulgarian Ministry of Education and Science.

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Draganov, B.R., Gadjev, I. Direct and Converse Voronovskaya Estimates for the Bernstein Operator. Results Math 73, 11 (2018). https://doi.org/10.1007/s00025-018-0795-8

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  • DOI: https://doi.org/10.1007/s00025-018-0795-8

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