Convergence of \(\lambda \)-Bernstein operators via power series summability method

Abstract

In this paper we present uniform convergence of a sequence of \(\lambda \) -Bernstein operators via A-statistical convergence and power summability method. A rate of convergence of the sequence of operators are also investigated by means of above mentioned summability methods. The last section is devoted to pointwise convergence (A-statistical convergence) of the sequence of operators in terms of Voronovskaya and Grü ss–Voronovskaya type theorems.

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References

  1. 1.

    Acar, T., Acu, A.M., Manav, N.: Approximation of functions by genuine Bernstein–Durrmeyer type operators. J. Math. Inequal. 12(4), 975–987 (2018)

    MathSciNet  Article  Google Scholar 

  2. 2.

    Acu, A.M., Manav, N., Sofonea, D.F.: Approximation properties of \(\lambda \)-Kantorovich operators. J. Inequal. Appl. 2018, 202 (2018)

    MathSciNet  Article  Google Scholar 

  3. 3.

    Anastassiou, G.A., Khan, M.A.: Korovkin type statistical approximation theorem for a function of two variables. J. Comput. Anal. Appl. 21(7), 1176–1184 (2016)

    MathSciNet  MATH  Google Scholar 

  4. 4.

    Atlihan, O.G., Unver, M., Duman, O.: Korovkin theorems on weighted spaces: revisited. Period. Math. Hungar. 75(2), 201–209 (2017)

    MathSciNet  Article  Google Scholar 

  5. 5.

    Basar, F.: Summability Theory and Its Applications. Bentham Science Publishers, Istanbul (2011)

    Google Scholar 

  6. 6.

    Boos, J.: Classical and Modern Methods in Summability. Oxford University Press, Oxford (2000)

    Google Scholar 

  7. 7.

    Braha, N.L.: Some weighted equi-statistical convergence and Korovkin type-theorem. Results Math. 70, 433–446 (2016)

    MathSciNet  Article  Google Scholar 

  8. 8.

    Braha, N.L., Loku, V., Srivastava, H.M.: \(\Lambda ^{2}\) -Weighted statistical convergence and Korovkin and Voronovskaya type theorems. Appl. Math. Comput. 266, 675–686 (2015)

    MathSciNet  MATH  Google Scholar 

  9. 9.

    Braha, B.L., Srivastava, H.M., Mohiuddine, S.A.: A Korovkin type approximation theorem for periodic functions via the summability of the modified de la Vallee Poussin mean. Appl. Math. Comput. 228, 162–169 (2014)

    MathSciNet  MATH  Google Scholar 

  10. 10.

    Braha, N.L.: Some properties of new modified Szász–Mirakyan operators in polynomial weight spaces via power summability methods. Bull. Math. Anal. Appl. 10(3), 53–65 (2018)

    MathSciNet  MATH  Google Scholar 

  11. 11.

    Braha, N.L.: Some properties of Baskakov–Schurer–Szász operators via power summability method. Quaest. Math. 42(10), 1411–1426 (2019)

    MathSciNet  Article  Google Scholar 

  12. 12.

    Cai, Q.-B., Lian, B.-Y., Zhou, G.: Approximation properties of \(\lambda \)-Bernstein operators. J. Inequal. Appl. 2018, 61 (2018)

    MathSciNet  Article  Google Scholar 

  13. 13.

    Cai, Q.-B.: The Bézier variant of Kantorovich type \( \lambda \)-Bernstein operators. J. Inequal. Appl. 2018, 90 (2018)

    Article  Google Scholar 

  14. 14.

    Campiti, M., Metafune, G.: \(L^{p}\)-convergence of Bernstein–Kantorovich-type operators. Ann. Polon. Math. 63(3), 273–280 (1996)

    MathSciNet  Article  Google Scholar 

  15. 15.

    Duman, O., Khan, M.K., Orhan, C.: \(A\)-Statistical convergence of approximating operators. Math. Inequal. Appl. 6, 689–699 (2003)

    MathSciNet  MATH  Google Scholar 

  16. 16.

    Fridy, J.A., Miller, H.I.: A matrix characterization of statistical convergence. Analysis 11, 59–66 (1991)

    MathSciNet  Article  Google Scholar 

  17. 17.

    Gal, S.G., Gonska, H.: Grüss and Grü ss-Voronovskaya-type estimates for some Bernstein-type polynomials of real and complex variables. Jaen J. Approx. 7(1), 97–122 (2015)

    MathSciNet  MATH  Google Scholar 

  18. 18.

    Kadak, U., Braha, N.L., Srivastava, H.M.: Statistical weighted \( {\cal{B}}\)-summability and its applications to approximation theorems. Appl. Math. Comput. 302, 80–96 (2017)

    MathSciNet  MATH  Google Scholar 

  19. 19.

    Karslı, H.: Approximation results for Urysohn type two dimensional nonlinear Bernstein operators. Constr. Math. Anal. 1(1), 45–57 (2018)

    Google Scholar 

  20. 20.

    Kratz, W., Stadtmuller, U.: Tauberian theorems for \(J_{p}\) -summability. J. Math. Anal. Appl. 139, 362–371 (1989)

    MathSciNet  Article  Google Scholar 

  21. 21.

    Loku, V., Braha, N.L.: Some weighted statistical convergence and Korovkin type-theorem. J. Inequal. Spec. Funct. 8(3), 139–150 (2017)

    MathSciNet  Google Scholar 

  22. 22.

    Mohiuddine, S.A., Alotaibi, A., Mursaleen, M.: Statistical summability \((C,1)\) and a Korovkin type approximation theorem. J. Ineq. Appl. 2012, 172 (2012)

    MathSciNet  Article  Google Scholar 

  23. 23.

    Mursaleen, M., Alotaibi, A.: Statistical summability and approximation by de la Vallée-Poussin mean. Appl. Math. Letters 24, 320–324 (2011)

    MathSciNet  Article  Google Scholar 

  24. 24.

    Mursaleen, M., Alotaibi, A.: Korovkin type approximation theorem for functions of two variables through statistical \(A\)-summability. Adv. Differ. Equ. 2012, 65 (2012)

    MathSciNet  Article  Google Scholar 

  25. 25.

    Mursaleen, M., Karakaya, V., Erturk, M., Gursoy, F.: Weighted statistical convergence and its application to Korovkin type approximation theorem. Appl. Math. Comput. 218, 9132–9137 (2012)

    MathSciNet  MATH  Google Scholar 

  26. 26.

    Mursaleen, M., Al-Abied, A.A.H., Salman, M.A.: Chlodowsky type \( (\lambda; q)\)-Bernstein–Stancu operators. Azerb. J. Math. 10(1), 75–101 (2020)

    MATH  Google Scholar 

  27. 27.

    Rahman, S., Mursaleen, M., Acu, A.M.: Approximation properties of \(\lambda \)-Bernstein–Kantorovich operators with shifted knots. Math. Methods Appl. Sci. 42(11), 4042–4053 (2019)

    MathSciNet  Article  Google Scholar 

  28. 28.

    Soylemez, D., Unver, M.: Korovkin Type Theorems for Cheney–Sharma Operators via Summability Methods. Results Math. 72(3), 1601–1612 (2017)

    MathSciNet  Article  Google Scholar 

  29. 29.

    Stadtmuller, U., Tali, A.: On certain families of generalized Nörlund methods and power series methods. J. Math. Anal. Appl. 238, 44–66 (1999)

    MathSciNet  Article  Google Scholar 

  30. 30.

    Tas, E., Yurdakadim, T.: Approximation by positive linear operators in modular spaces by power series method. Positivity 21(4), 1293–1306 (2017)

    MathSciNet  Article  Google Scholar 

  31. 31.

    Tas, E.: Some results concerning Mastroianni operators by power series method. Commun. Fac. Sci. Univ. Ank. Ser. A1 Math. Stat. 63(1), 187–195 (2016)

    MathSciNet  MATH  Google Scholar 

  32. 32.

    Unver, M.: Abel transforms of positive linear operators. In: ICNAAM 2013. AIP Conference Proceedings, vol. 1558, pp. 1148–1151 (2013)

  33. 33.

    Ye, Z., Long, X., Zeng, X.-M.: Adjustment algorithms for Bézier curve and surface. In: International Conference on Computer Science and Education, pp. 1712–1716 (2010)

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Correspondence to M. Mursaleen.

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Braha, N.L., Mansour, T., Mursaleen, M. et al. Convergence of \(\lambda \)-Bernstein operators via power series summability method. J. Appl. Math. Comput. (2020). https://doi.org/10.1007/s12190-020-01384-x

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Keywords

  • A-statistical convergence
  • \(\lambda \)-Bernstein operators
  • Power summability method
  • Korovkin type theorem
  • Voronovskaya type theorem
  • Rate of convergence
  • Grüss–Voronovskaya type theorem

Mathematics Subject Classification

  • 40G10
  • 40C15
  • 41A36
  • 40A35