Ukrainian Mathematical Journal

, Volume 66, Issue 6, pp 928–936 | Cite as

Some Approximation Properties of Szasz–Mirakyan–Bernstein Operators of the Chlodovsky Type

  • T. Tunc
  • E. Simsek

We motivate a new sequence of positive linear operators by means of the Chlodovsky-type Szasz–Mirakyan–Bernstein operators and investigate some approximation properties of these operators in the space of continuous functions defined on the right semiaxis. We also find the order of this approximation by using the modulus of continuity and present the Voronovskaya-type theorem.


Approximation Property Bernstein Polynomial Positive Linear Operator Continuity Point Total Positivity 
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  1. 1.
    S. N. Bernstein, “Démonstration du théorème de Weierstrass Fondée sur le Calcul des Probabilités,” Comm. Soc. Math. Kharkow, 13, No. 2, 1–2 (1912–1913).Google Scholar
  2. 2.
    G. G. Lorentz, Bernstein Polynomials, Chelsea Publ., New York (1986).zbMATHGoogle Scholar
  3. 3.
    I. Chlodovsky, “Sur le développement des fonctions définies dans un interval infini en séries de polynomes de M. S. Bernstein,” Compos. Math., 4, 380–393 (1937).MathSciNetGoogle Scholar
  4. 4.
    O. Szasz, “Generalizations of S. Bernstein’s polynomials to the infinite interval,” J. Research Nat. Bureau Standards, 45, No. 3, 239–245 (1950).CrossRefMathSciNetGoogle Scholar
  5. 5.
    E. Ibikli, “Approximation by Bernstein–Chlodovsky polynomials,” Hacettepe J. Math. Statist., 32, 1–5 (2003).MathSciNetzbMATHGoogle Scholar
  6. 6.
    H. Karsli, “A Voronovskaya-type theorem for the second derivative of the Bernstein–Chlodovsky polynomials,” Proc. Eston. Acad. Sci. Math., 61, No. 1, 9–19 (2012).CrossRefMathSciNetzbMATHGoogle Scholar
  7. 7.
    D. D. Stancu, “Approximation of functions by a new class of linear polynomial operator,” Rev. Roum. Math. Pures Appl., 13, 1173–1194 (1968).MathSciNetzbMATHGoogle Scholar
  8. 8.
    G. M. Phillips, “Bernstein polynomials based on the q-integers,” Ann. Number. Math., 4, 511–518 (1997).zbMATHGoogle Scholar
  9. 9.
    S. Ostrovska, “On the limit q-Bernstein operators,” Math. Balkan, 18, 165–172 (2004).MathSciNetzbMATHGoogle Scholar
  10. 10.
    G. M. Mirakyan, “Approximation of continuous functions with the aid of polynomials of the form e -nxΣk=0 mn C k,n x k ,Akad. Nauk SSSR, 31, 201–205 (1941).Google Scholar
  11. 11.
    J. Favard, “Sur les multiplicateurs d’interpolation,” J. Math. Pures Appl., 23, No. 9, 219–247 (1944).MathSciNetzbMATHGoogle Scholar
  12. 12.
    J. Albrycht and J. Radecki, “On a generalization of the theorem of Voronovskaya,” Zesz. Nauk. UAM, 2, 1–7 (1960).Google Scholar

Copyright information

© Springer Science+Business Media New York 2014

Authors and Affiliations

  • T. Tunc
    • 1
  • E. Simsek
    • 1
  1. 1.Mersin UniversityMersinTurkey

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