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Mathematica Slovaca

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Volume 66, Issue 3


Weighted αβ-statistical convergence of Kantorovich-Mittag-Leffler operators

Mehmet Ali özarslan / Hüseyin Aktuğlu
Published Online: 2016-08-26 | DOI: https://doi.org/10.1515/ms-2015-0171


In this paper we introduce Kantorovich variant of the Mittag-Leffler operators including the modified Kantorovich-Szász-Mirakjan operators. We give αβ-statistical approximation theorems for these operators in various function spaces. The results include the statistical, lacunary statistical and λ-statistical cases. Moreover, we compute the rate of convergence in different Lipschitz type spaces.

MSC 2010: Primary 41A25, 41A35, 40G99; Secondary 47A58

Keywords: Kantorovich-Mittag-Leffler operators; Szász-Mirakjan operators; αβ-statistical convergence; λ-statistical convergence; statistical convergence; Lacunary statistical convergence; Lipschitz class functionals; Bernoulli numbers


  • [1]

    Aktuglu, H.: Korovkin type approximation theorems proved via αβ-statistical convergence, J. Comput. Appl. Math. 259 (2014), 174–181.Google Scholar

  • [2]

    Aktuglu, H.—Gezer, H.: Lacunary equi-statistical convergence of positive linear operators, Cent. Eur. J. Math. 7 (2009), 558–567.Google Scholar

  • [3]

    Aktuglu, H.—Ozarslan, M. A.—Gezer, H.: A-Equistatistical Convergence of Positive Linear Operators, J. Comput. Appl. Math. 12 (2010), 24–36.Google Scholar

  • [4]

    Altin, A.—Doğru, O.—Tasdelen, F.: The generalization of Meyer-Konig and Zeller operators by generating functions, J. Math. Anal. Appl. 312 (2005), 181–194.Google Scholar

  • [5]

    Aral, A.—Duman, O.: A Voronovskaya-type formula for SMK operators via statistical convergence, Math. Slovaca 61 (2011), 235–244.Google Scholar

  • [6]

    Bretti, G.—Natalini, P.—Ricci, P. E.: Generalizations of the Bernoulli and Appell polynomials, Abstr. Appl. Anal. (2004), No. 7, 613–623.Google Scholar

  • [7]

    Doğru, O.—Orkcu, M.: Statistical approximation by a modification of q-Meyer-König and Zeller operators, Appl. Math. Lett. 23 (2010), 261–266.Google Scholar

  • [8]

    Doğru, O.—Ozarslan, M. A.—Tasdelen, F.: On positive operators involving a certain class of generating functions, Studia Sci. Math. Hungar. 41 (2004), 415–429.Google Scholar

  • [9]

    Duman, O.: A-statistical convergence of sequences of convolution operators, Taiwanese J. Math. 12 (2008), 523–536.Google Scholar

  • [10]

    Duman, O.—Orhan, C.: Rates of A-statistical convergence of positive linear operators, Appl. Math. Lett. 18 (2005), 1339–1344.Google Scholar

  • [11]

    Duman, O.—Orhan, C.: Rates of A-statistical convergence of operators in the space of locally integrable functions, Appl. Math. Lett. 21 (2008), 431–435.Google Scholar

  • [12]

    Duman, O.—Orhan, C.: Statistical approximation by positive linear operators, Studia Math. 161 (2004), 187–197.Google Scholar

  • [13]

    Erkus, E.—Duman, O.—Srivastava, H. M.: Statistical approximation of certain positive linear operators constructed by means of the Chan-Chyan-Srivastava polynomials, Appl. Math. Comput. 182 (2006), 213–222.Google Scholar

  • [14]

    Erkus-Duman, E.—Duman, O.: Statistical approximation properties of high order operators constructed with the Chan-Chyan-Srivastava polynomials, Appl. Math. Comput. 218 (2011), 1927–1933.Google Scholar

  • [15]

    Fast, H.: Sur la convergence statistique, Collog. Math. 2 (1951), 241–244.Google Scholar

  • [16]

    Freedman, A. R.—Sember, J. J.: Densities and summability, Pacific J. Math. 95 (1981), 293–305.Google Scholar

  • [17]

    Fridy, J. A.: On stastistical convergence, Analysis (Munich) 5 (1985), 301–313.Google Scholar

  • [18]

    Gadjiev, A. D.—Orhan, C.: Some approximation theorems via statistical convergence, Rocky Mountain J. Math. 32 (2002), 129–138.Google Scholar

  • [19]

    Gupta, V.—Naokant, D.: A note on improved estimations for integrated Szász-Mirakyan operators, Math. Slovaca 61 (2011), 799–806.Google Scholar

  • [20]

    Kolk, E.: Matrix summability of statistically convergent sequences, Analysis (Munich) 13 (1993), 77–83.Google Scholar

  • [21]

    Mahmudov, N. I.: q-Szász operators which preserve x2, Math. Slovaca 63 (2013), 1059–1072.Web of ScienceGoogle Scholar

  • [22]

    Miller, H. I.: A measure theoretical subsequence characterization of statistical convergence, Trans. Amer. Math. Soc. 347 (1995), 1811–1819.Google Scholar

  • [23]

    Mittag-Leffler, G. M.: Sur la nouvelle function Eα, C. R. Math. Acad. Sci. Paris 137 (1903), 554–558.Google Scholar

  • [24]

    Ozarslan, M. A.: A-statistical convergence of Mittag-Leffler Operators, Miskolc Math. Notes 14 (2013), 209–217.Google Scholar

  • [25]

    Ozarslan, M. A.: q-Laguerre type linear positive operators, Studia Sci. Math. Hungar. 44 (2007), 65–80.Google Scholar

  • [26]

    Ozarslan, M. A.—Aktuglu, H.: A-statistical approximation of generalized Szász-Mirakjan-Beta operators, Appl. Math. Lett. 24 (2011), 1785–1790.Google Scholar

  • [27]

    Ozarslan, M. A.—Duman, O.: Approximation properties of Poisson integrals for orthogonal expansions, Taiwanese J. Math. 12 (2008), 1147–1163.Google Scholar

  • [28]

    Ozarslan, M. A.—Duman, O.—Srivastava, H. M.: Statistical approximation results for Kantorovich-type operators involving some special polynomials, Math. Comput. Modelling 48 (2008), 388–401.Google Scholar

  • [29]

    Rempulska, L.—Graczyk, S.: On generalized Szász-Mirakyan operators of functions of two variables, Math. Slovaca 62 (2012), 87–98.Google Scholar

  • [30]

    Steinhaus, H.: Sur la convergence ordinaire et la convergence asympttique, Colloq. Math. 2 (1951), 73–74.Google Scholar

  • [31]

    Szasz, O.: Generalization of S. Bernstein’s polynomials to the infinite interval, J. Res. Nat. Bur. Standards 45 (1950), 239–245.Google Scholar

  • [32]

    Tasdelen, F.—Aktas, R.—Altin, A.: A Kantorovich Type of Szasz Operators Including Brenke-Type Polynomials, Abstr. Appl. Anal. (2012), Article No. 867203, DOI: .CrossrefGoogle Scholar

  • [33]

    Tasdelen, F.—Erencin, A.: The generalization of bivariate MKZ operators by multiple generating functions, J. Math. Anal. Appl. 331 (2007), 727–735.Google Scholar

  • [34]

    Varma, S.—Icoz, G.—Sucu, S.: On Some Extensions of Szasz Operators Including Boas-Buck-Type Polynomials, Abstr. Appl. Anal. (2012), Article No. 680340, DOI: .CrossrefGoogle Scholar

  • [35]

    Varma, S.—Sucu, S.—Icoz, G.: Generalization of Szasz operators involving Brenke type polynomials, Comput. Math. Appl. 64 (2012), 121–127.Google Scholar

  • [36]

    Varma, S.—Tasdelen, F.: Szasz type operators involving Charlier polynomials, Math. Comput. Modelling 56 (2012), 118–122.Google Scholar

  • [37]

    Walczak, Z.: Error estimates and the Voronovskaja theorem for modified Szász-Mirakyan operators, Math. Slovaca 55 (2005), 465–476.Google Scholar

  • [38]

    Wiman, A.: Über den Fundamentalsatz in der Teorie der Funktionen Eα(x), Acta Math. 29 (1905), 191–201 (German).CrossrefGoogle Scholar

About the article

Received: 2013-07-01

Accepted: 2014-01-18

Published Online: 2016-08-26

Published in Print: 2016-06-01

Citation Information: Mathematica Slovaca, Volume 66, Issue 3, Pages 695–706, ISSN (Online) 1337-2211, ISSN (Print) 0139-9918, DOI: https://doi.org/10.1515/ms-2015-0171.

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