Numerical Algorithms

, Volume 77, Issue 1, pp 111–150 | Cite as

A new approach to improve the order of approximation of the Bernstein operators: theory and applications

  • Hassan Khosravian-Arab
  • Mehdi DehghanEmail author
  • M. R. Eslahchi
Original Paper


This paper presents a new approach to improve the order of approximation of the Bernstein operators. Three new operators of the Bernstein-type with the degree of approximations one, two, and three are obtained. Also, some theoretical results concerning the rate of convergence of the new operators are proved. Finally, some applications of the obtained operators such as approximation of functions and some new quadrature rules are introduced and the theoretical results are verified numerically.


Linear positive operator The Bernstein operators Bernstein Theorem Korovkin Theorem Voronovkaja Theorem Rate of convergence Degree of convergence 

Mathematics Subject Classification (2010)

41A10 42A20 65D30 


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The authors would like to thank the anonymous reviewers for their careful reading of the manuscript and their comments which improved the quality of the paper.


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Copyright information

© Springer Science+Business Media New York 2017

Authors and Affiliations

  • Hassan Khosravian-Arab
    • 1
  • Mehdi Dehghan
    • 1
    Email author
  • M. R. Eslahchi
    • 2
  1. 1.Department of Applied Mathematics, Faculty of Mathematics Computer SciencesAmirkabir University of TechnologyTehranIran
  2. 2.Department of Applied Mathematics, Faculty of Mathematical SciencesTarbiat Modares UniversityTehranIran

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