The Limit q-Bernstein Operators with Varying q

  • Manal Mastafa Almesbahi
  • Sofiya OstrovskaEmail author
  • Mehmet Turan
Part of the Nonlinear Systems and Complexity book series (NSCH, volume 23)


In this paper, the continuity of the limit q-Bernstein operator with respect to parameter q is investigated. It is proved that the map qBq is continuous in the strong operator topology on C[0, 1] for q ∈ [0, 1]. Meanwhile, in the uniform operator topology, this map is discontinuous at every q ∈ [0, 1].


  1. 1.
    Andrews, G.E., Askey, R. Roy, R.: Special Functions. Cambridge University Press. Cambridge (1999)CrossRefGoogle Scholar
  2. 2.
    Bernstein, S.N.: Démonstration du théorème de Weierstrass fondée sur la calcul des probabilités. Commun. Soc. Math. Charkow série 2, 13, 1–2 (1912)zbMATHGoogle Scholar
  3. 3.
    Charalambides, Ch.A.: Discrete q-Distributions. Wiley, New York (2016)CrossRefGoogle Scholar
  4. 4.
    Il’inskii, A.: A probabilistic approach to q-polynomial coefficients, Euler and Stirling numbers I. Matematicheskaya Fizika, Analiz Geometriya 11(4), 434–448 (2004)MathSciNetzbMATHGoogle Scholar
  5. 5.
    Il’inskii, A. Ostrovska, S.: Convergence of generalized Bernstein polynomials. J. Approx. Theory 116(1), 100–112 (2002)MathSciNetCrossRefGoogle Scholar
  6. 6.
    Kreyszig, E.: Introductory Functional Analysis with Applications. Wiley, New York (1978)zbMATHGoogle Scholar
  7. 7.
    Ostrovska, S.: A survey of results on the limit q-Bernstein operator. J. Appl. Math. 2013, Article ID 159720, 7 pp. (2013)Google Scholar
  8. 8.
    Ostrovska, S.: The q-versions of the Bernstein operator: from mere analogies to further developments. Results Math. 69(3), 275–295 (2016)MathSciNetCrossRefGoogle Scholar
  9. 9.
    Phillips, G.M.: Bernstein polynomials based on the q-integers. Ann. Numer. Math. 4, 511–518 (1997)MathSciNetzbMATHGoogle Scholar
  10. 10.
    Phillips, G.M.: A survey of results on the q-Bernstein polynomials. IMA J. Numer. Anal. 30, 277–288 (2010)MathSciNetCrossRefGoogle Scholar
  11. 11.
    Videnskii, V.S.: On some classes of q-parametric positive operators. Oper. Theory Adv. Appl. 158, 213–222 (2005)MathSciNetCrossRefGoogle Scholar

Copyright information

© Springer International Publishing AG, part of Springer Nature 2019

Authors and Affiliations

  • Manal Mastafa Almesbahi
    • 1
  • Sofiya Ostrovska
    • 1
    Email author
  • Mehmet Turan
    • 1
  1. 1.Atilim UniversityDepartment of MathematicsAnkaraTurkey

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