, Volume 23, Issue 1, pp 1–10 | Cite as

Statistical equi-equal convergence of positive linear operators

  • Fadime DirikEmail author
  • Pınar Okçu Şahin


Many researchers have been interested in the concept of statistical convergence because of the fact that it is stronger than the classical convergence. Also, the concepts of statistical equal convergence and equi-statistical convergence are more general than the statistical uniform convergence. In this paper we define a new type of statistical convergence by using the notions of equi-statistical convergence and statistical equal convergence to prove a Korovkin type theorem. We show that our theorem is a non-trivial extension of some well-known Korovkin type approximation theorems which were demonstrated by earlier authors. After, we present an example in support of our definition and result presented in this paper. Finally, we also compute the rates of statistical equi-equal convergence of sequences of positive linear operators.


Statistical equal convergence Equi-statistical convergence Positive linear operators Korovkin theorem Modulus of continuity 

Mathematics Subject Classification

41A25 41A36 


  1. 1.
    Acar, T., Dirik, F.: Korovkin-type theorems in weighted Lp-spaces via summation process. Sci. World J. ArticleID 534054(2013)Google Scholar
  2. 2.
    Altomare, F., Campiti, M.: Korovkin-Type Approximation Theory and Its Applications. De Gruyter Stud. Math. Walter de Gruyter, Berlin (1994)CrossRefzbMATHGoogle Scholar
  3. 3.
    Balcerzak, M., Dems, K., Komisarski, A.: Statistical convergence and ideal convergence for sequences of functions. J. Math. Anal. Appl. 328, 715–729 (2007)MathSciNetCrossRefzbMATHGoogle Scholar
  4. 4.
    Császár, A., Laczkovich, M.: Discrete and equal convergence. Studia Sci. Math. Hungar. 10(3–4), 463–472 (1975)MathSciNetzbMATHGoogle Scholar
  5. 5.
    Das, P., Dutta, S., Pal, S.K.: On \(I\) and \(I^{\ast }\)- equal convergence and an Egoroff-type theorem. Mat. Vesnik 66(2), 165–177 (2014)MathSciNetzbMATHGoogle Scholar
  6. 6.
    Császár, A., Laczkovich, M.: Some remarks on discrete Baire classes. Acta Math. Acad. Sci. Hung. 33(1–2), 51–70 (1979)MathSciNetCrossRefzbMATHGoogle Scholar
  7. 7.
    Demirci, K., Dirik, F.: Statistical entension of the Korovkin-type approximation theorem. Appl. Math. E-Notes 11, 101–109 (2011)MathSciNetGoogle Scholar
  8. 8.
    Demirci, K., Orhan, S.: Statistically relatively uniform convergence of positive linear operators. Results Math. 69(3–4), 359–367 (2016)MathSciNetCrossRefzbMATHGoogle Scholar
  9. 9.
    Dirik, F., Demirci, K.: Korovkin type approximation theorem for functions of two variables in statistical sense. Turk. J. Math. 34, 73–83 (2010)MathSciNetzbMATHGoogle Scholar
  10. 10.
    Duman, O., Orhan, C.: \(\mu \)-statistically convergent function sequences. Czechoslovak Math. J. 54, 413–422 (2004)MathSciNetCrossRefzbMATHGoogle Scholar
  11. 11.
    Fast, H.: Sur la convergence statistique. Colloq. Math. 2, 241–244 (1951)MathSciNetCrossRefzbMATHGoogle Scholar
  12. 12.
    Gadjiev, A.D., Orhan, C.: Some approximation theorems via statistical convergence. Rocky Mt. J. Math. 32, 129–138 (2002)MathSciNetCrossRefzbMATHGoogle Scholar
  13. 13.
    Karakuş, S., Demirci, K., Duman, O.: Equi-statistical convergence of positive linear operators. J. Math. Anal. Appl. 339, 1065–1072 (2008)MathSciNetCrossRefzbMATHGoogle Scholar
  14. 14.
    Korovkin, P.P.: Linear Operators and Approximation Theory. Hindustan Publ. Co., Delhi (1960)Google Scholar
  15. 15.
    Okçu Şahin, P., Dirik, F.: A Korovkin-type theorem for double sequences of positive linear operators via power series method. Positivity 22(1), 209–218 (2008)MathSciNetzbMATHGoogle Scholar
  16. 16.
    Ozguc, I., Tas, E.: A Korovkin-type approximation theorem and power series method. Results Math. 69, 497–504 (2016)MathSciNetCrossRefzbMATHGoogle Scholar
  17. 17.
    Steinhaus, H.: Sur la convergence ordinaire et la convergence asymtotique. Colloq. Math. 2, 73–74 (1951)CrossRefGoogle Scholar

Copyright information

© Springer Nature Switzerland AG 2018

Authors and Affiliations

  1. 1.Department of MathematicsSinop UniversitySinopTurkey

Personalised recommendations