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A Survey of Results on the Limit -Bernstein Operator

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  • Sofiya Ostrovska

Abstract

The limit -Bernstein operator emerges naturally as a modification of the Szász-Mirakyan operator related to the Euler distribution, which is used in the -boson theory to describe the energy distribution in a -analogue of the coherent state. At the same time, this operator bears a significant role in the approximation theory as an exemplary model for the study of the convergence of the -operators. Over the past years, the limit -Bernstein operator has been studied widely from different perspectives. It has been shown that is a positive shape-preserving linear operator on with . Its approximation properties, probabilistic interpretation, the behavior of iterates, and the impact on the smoothness of a function have already been examined. In this paper, we present a review of the results on the limit -Bernstein operator related to the approximation theory. A complete bibliography is supplied.

Suggested Citation

  • Sofiya Ostrovska, 2013. "A Survey of Results on the Limit -Bernstein Operator," Journal of Applied Mathematics, Hindawi, vol. 2013, pages 1-7, March.
    • Handle: RePEc:hin:jnljam:159720
    DOI: 10.1155/2013/159720
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