Home Mathematics Weighted αβ-statistical convergence of Kantorovich-Mittag-Leffler operators
Article
Licensed
Unlicensed Requires Authentication

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

  • Mehmet Ali özarslan EMAIL logo and Hüseyin Aktuğlu
Published/Copyright: August 26, 2016
Become an author with De Gruyter Brill
Author Information Explore this Subject
Mathematica Slovaca
From the journal Mathematica Slovaca Volume 66 Issue 3

Abstract

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

(Communicated by Gregor Dolinar)

References

[1] Aktuglu, H.: Korovkin type approximation theorems proved via αβ-statistical convergence, J. Comput. Appl. Math. 259 (2014), 174–181.10.1016/j.cam.2013.05.012Search in Google Scholar

[2] Aktuglu, H.—Gezer, H.: Lacunary equi-statistical convergence of positive linear operators, Cent. Eur. J. Math. 7 (2009), 558–567.10.2478/s11533-009-0009-4Search in 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.Search in 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), 181–194.10.1016/j.jmaa.2005.03.086Search in Google Scholar

[5] Aral, A.—Duman, O.: A Voronovskaya-type formula for SMK operators via statistical convergence, Math. Slovaca 61 (2011), 235–244.10.2478/s12175-011-0008-3Search in Google Scholar

[6] Bretti, G.—Natalini, P.—Ricci, P. E.: Generalizations of the Bernoulli and Appell polynomials, Abstr. Appl. Anal. (2004), No. 7, 613–623.10.1155/S1085337504306263Search in Google Scholar

[7] Doğru, O.—Orkcu, M.: Statistical approximation by a modification of q-Meyer-König and Zeller operators, Appl. Math. Lett. 23 (2010), 261–266.10.1016/j.aml.2009.09.018Search in 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), 415–429.10.1556/sscmath.41.2004.4.5Search in Google Scholar

[9] Duman, O.: A-statistical convergence of sequences of convolution operators, Taiwanese J. Math. 12 (2008), 523–536.10.11650/twjm/1500574172Search in Google Scholar

[10] Duman, O.—Orhan, C.: Rates of A-statistical convergence of positive linear operators, Appl. Math. Lett. 18 (2005), 1339–1344.10.1016/j.aml.2005.02.029Search in 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), 431–435.10.1016/j.aml.2007.05.017Search in Google Scholar

[12] Duman, O.—Orhan, C.: Statistical approximation by positive linear operators, Studia Math. 161 (2004), 187–197.10.4064/sm161-2-6Search in 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), 213–222.10.1016/j.amc.2006.01.090Search in 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), 1927–1933.10.1016/j.amc.2011.07.004Search in Google Scholar

[15] Fast, H.: Sur la convergence statistique, Collog. Math. 2 (1951), 241–244.10.4064/cm-2-3-4-241-244Search in Google Scholar

[16] Freedman, A. R.—Sember, J. J.: Densities and summability, Pacific J. Math. 95 (1981), 293–305.10.2140/pjm.1981.95.293Search in Google Scholar

[17] Fridy, J. A.: On stastistical convergence, Analysis (Munich) 5 (1985), 301–313.10.1524/anly.1985.5.4.301Search in Google Scholar

[18] Gadjiev, A. D.—Orhan, C.: Some approximation theorems via statistical convergence, Rocky Mountain J. Math. 32 (2002), 129–138.10.1216/rmjm/1030539612Search in Google Scholar

[19] Gupta, V.—Naokant, D.: A note on improved estimations for integrated Szász-Mirakyan operators, Math. Slovaca 61 (2011), 799–806.10.2478/s12175-011-0047-9Search in Google Scholar

[20] Kolk, E.: Matrix summability of statistically convergent sequences, Analysis (Munich) 13 (1993), 77–83.10.1524/anly.1993.13.12.77Search in Google Scholar

[21] Mahmudov, N. I.: q-Szász operators which preserve x2, Math. Slovaca 63 (2013), 1059–1072.10.1016/j.cam.2010.03.031Search in Google Scholar

[22] Miller, H. I.: A measure theoretical subsequence characterization of statistical convergence, Trans. Amer. Math. Soc. 347 (1995), 1811–1819.10.1090/S0002-9947-1995-1260176-6Search in Google Scholar

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

[24] Ozarslan, M. A.: A-statistical convergence of Mittag-Leffler Operators, Miskolc Math. Notes 14 (2013), 209–217.10.18514/MMN.2013.475Search in Google Scholar

[25] Ozarslan, M. A.: q-Laguerre type linear positive operators, Studia Sci. Math. Hungar. 44 (2007), 65–80.10.1556/sscmath.44.2007.1.7Search in Google Scholar

[26] Ozarslan, M. A.—Aktuglu, H.: A-statistical approximation of generalized Szász-Mirakjan-Beta operators, Appl. Math. Lett. 24 (2011), 1785–1790.10.1016/j.aml.2011.04.032Search in Google Scholar

[27] Ozarslan, M. A.—Duman, O.: Approximation properties of Poisson integrals for orthogonal expansions, Taiwanese J. Math. 12 (2008), 1147–1163.10.11650/twjm/1500574254Search in 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), 388–401.10.1016/j.mcm.2007.08.015Search in Google Scholar

[29] Rempulska, L.—Graczyk, S.: On generalized Szász-Mirakyan operators of functions of two variables, Math. Slovaca 62 (2012), 87–98.10.2478/s12175-011-0074-6Search in Google Scholar

[30] Steinhaus, H.: Sur la convergence ordinaire et la convergence asympttique, Colloq. Math. 2 (1951), 73–74.10.4064/cm-2-2-98-108Search in Google Scholar

[31] Szasz, O.: Generalization of S. Bernstein’s polynomials to the infinite interval, J. Res. Nat. Bur. Standards 45 (1950), 239–245.10.6028/jres.045.024Search in 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: 10.1155/2012/867203.Search in Google Scholar

[33] Tasdelen, F.—Erencin, A.: The generalization of bivariate MKZ operators by multiple generating functions, J. Math. Anal. Appl. 331 (2007), 727–735.10.1016/j.jmaa.2006.09.024Search in 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: 10.1155/2012/680340.Search in Google Scholar

[35] Varma, S.—Sucu, S.—Icoz, G.: Generalization of Szasz operators involving Brenke type polynomials, Comput. Math. Appl. 64 (2012), 121–127.10.1016/j.camwa.2012.01.025Search in Google Scholar

[36] Varma, S.—Tasdelen, F.: Szasz type operators involving Charlier polynomials, Math. Comput. Modelling 56 (2012), 118–122.10.1016/j.mcm.2011.12.017Search in Google Scholar

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

[38] Wiman, A.: Über den Fundamentalsatz in der Teorie der Funktionen Eα(x), Acta Math. 29 (1905), 191–201 (German).10.1007/BF02403202Search in Google Scholar

Received: 2013-7-1
Accepted: 2014-1-18
Published Online: 2016-8-26
Published in Print: 2016-6-1

© 2016 Mathematical Institute Slovak Academy of Sciences

Articles in the same Issue

  1. Research Article
  2. On the vertex-to-edge duality between the Cayley graph and the coset geometry of von Dyck groups
  3. Research Article
  4. Characterization of hereditarily reversible posets
  5. Research Article
  6. Families of sets which can be represented as sublattices of the lattice of convex subsets of a linearly ordered set
  7. Research Article
  8. Diophantine equation X4+Y4 = 2(U4 + V4)
  9. Research Article
  10. On the congruent number problem over integers of cyclic number fields
  11. Research Article
  12. On the Diophantine equation x2 + C= yn for C = 2a3b17c and C = 2a13b17c
  13. Research Article
  14. Representations and evaluations of the error term in a certain divisor problem
  15. Research Article
  16. A construction of a Peano curve
  17. Research Article
  18. Compositions of ϱ-upper continuous functions
  19. Research Article
  20. Products of Świa̧tkowski functions
  21. Research Article
  22. Subclasses of meromorphically p-valent functions involving a certain linear operator
  23. Research Article
  24. Oscillation results for second-order mixed neutral differential equations with distributed deviating arguments
  25. Research Article
  26. Oscillation criteria for higher order nonlinear delay dynamic equations on time scales
  27. Research Article
  28. On the solutions of a fourth order parabolic equation modeling epitaxial thin film growth
  29. Research Article
  30. Meromorphic solutions of q-shift difference equations
  31. Research Article
  32. Oscillation criteria for higher-order nonlinear delay dynamic equations on time scales
  33. Research Article
  34. Multiplier spaces and the summing operator for series
  35. Research Article
  36. Weighted αβ-statistical convergence of Kantorovich-Mittag-Leffler operators
  37. Research Article
  38. The local spectral radius of a nonnegative orbit of compact linear operators
  39. Research Article
  40. A note on Lie product preserving maps on Mn(ℝ)
  41. Research Article
  42. On vector optimality conditions for constrained problems with ℓ-stable data
  43. Research Article
  44. Reparameterization of weakly nonlinear regression models with constraints
  45. Research Article
  46. Convolutions of polynomial kernels
  47. Research Article
  48. Closed form evaluation of sums containing squares of Fibonomial coefficients
https://doi.org/10.1515/ms-2015-0171
Keywords for this article
Kantorovich-Mittag-Leffler operators; Szász-Mirakjan operators; αβ-statistical convergence; λ-statistical convergence; statistical convergence; Lacunary statistical convergence; Lipschitz class functionals; Bernoulli numbers

Articles in the same Issue

  1. Research Article
  2. On the vertex-to-edge duality between the Cayley graph and the coset geometry of von Dyck groups
  3. Research Article
  4. Characterization of hereditarily reversible posets
  5. Research Article
  6. Families of sets which can be represented as sublattices of the lattice of convex subsets of a linearly ordered set
  7. Research Article
  8. Diophantine equation X4+Y4 = 2(U4 + V4)
  9. Research Article
  10. On the congruent number problem over integers of cyclic number fields
  11. Research Article
  12. On the Diophantine equation x2 + C= yn for C = 2a3b17c and C = 2a13b17c
  13. Research Article
  14. Representations and evaluations of the error term in a certain divisor problem
  15. Research Article
  16. A construction of a Peano curve
  17. Research Article
  18. Compositions of ϱ-upper continuous functions
  19. Research Article
  20. Products of Świa̧tkowski functions
  21. Research Article
  22. Subclasses of meromorphically p-valent functions involving a certain linear operator
  23. Research Article
  24. Oscillation results for second-order mixed neutral differential equations with distributed deviating arguments
  25. Research Article
  26. Oscillation criteria for higher order nonlinear delay dynamic equations on time scales
  27. Research Article
  28. On the solutions of a fourth order parabolic equation modeling epitaxial thin film growth
  29. Research Article
  30. Meromorphic solutions of q-shift difference equations
  31. Research Article
  32. Oscillation criteria for higher-order nonlinear delay dynamic equations on time scales
  33. Research Article
  34. Multiplier spaces and the summing operator for series
  35. Research Article
  36. Weighted αβ-statistical convergence of Kantorovich-Mittag-Leffler operators
  37. Research Article
  38. The local spectral radius of a nonnegative orbit of compact linear operators
  39. Research Article
  40. A note on Lie product preserving maps on Mn(ℝ)
  41. Research Article
  42. On vector optimality conditions for constrained problems with ℓ-stable data
  43. Research Article
  44. Reparameterization of weakly nonlinear regression models with constraints
  45. Research Article
  46. Convolutions of polynomial kernels
  47. Research Article
  48. Closed form evaluation of sums containing squares of Fibonomial coefficients
Sign up now to receive a 20% welcome discount
Institutional Access
How does access work?
Have an idea on how to improve our website?
Please write us.
© 2025 De Gruyter Brill
Downloaded on 16.12.2025 from https://www.degruyterbrill.com/document/doi/10.1515/ms-2015-0171/pdf
Scroll to top button