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Year 2025, Volume: 13 Issue: 2, 84 - 91, 26.06.2025

Tuğba Değirmenci , Merve Ilkhan Kara

https://doi.org/10.36753/mathenot.1657527

Abstract

References

  • [1] Fast, H.: Sur la convergence statistique. Colloquium Mathematicum. 2, 241–244 (1951).
  • [2] Fridy, J. A., Miller, H. I.: A matrix characterization of statistical convertence. Analysis 11. 11(1), 59-66 (1991).
  • [3] Braha, N. L.: Some properties of Baskakov–Schurer-Szász operators via power summability method. Quaestiones Mathematicae. 42(10), 1411-1426 (2019).
  • [4] Kratz, W., Stadtmuller, U.: Tauberian theorems for Jp-summability. Journal of Mathematical Analysis and Applications 139, 362–371 (1989).
  • [5] Stadtmuller, U., Tali, A.: On certain families of generalized Nörlund methods and power series methods. Journal of Mathematical Analysis and Applications 238(1), 44–66 (1999).
  • [6] Başar, F.: Summability theory and its applications. Bentham Science Publishers. (2012).
  • [7] Braha, N. L.: Some weighted equi-statistical convergence and Korovkin type-theorem. Results in Mathematics. 70(3–4), 433–446 (2016).
  • [8] Taş, E.: Some results concerning Mastroianni operators by power series method. Communications Faculty of Sciences University of Ankara Series A1 Mathematics and Statistics. 63(1), 187-195 (2016).
  • [9] Taş, E., Yurdakadim, T.: Approximation by positive linear operators in modular spaces by power series method. Positivity. 21(4), 1293-1306 (2017).
  • [10] Boos, J.: Classical and modern methods in summability. Oxford University Press. Oxford (2000).
  • [11] Campiti, M., Metafune, G.: L Lp-convergence of Bernstein–Kantorovich-type operators. Annales Polonici Mathematici. 63(3), 273-280 (1996).
  • [12] Braha, N. L., Loku, V., Srivastava, H. M.: Λ2-Weighted statistical convergence and Korovkin and Voronovskaya type theorems. Applied Mathematics and Computation. 266, 675-686 (2015).
  • [13] Braha, N.L.: Some properties of new modified Szász–Mirakyan operators in polynomial weight spaces via power summability method. Bulletin of Mathematical Analysis and Applications. 10(3), 53-65 (2018).
  • [14] Uğur, K., Braha, N.L., Srivastava, H. M.: Statistical weighted B-summability and its applications to approximation theorems. Applied Mathematics and Computation. 302, 80–96 (2017).
  • [15] Loku, V., Braha, N. L.: Some weighted statistical convergence and Korovkin type-theorem. Journal of Inequalities and Special Functions. 8(3), 139–150 (2017).
  • [16] Usta, F: On new modification of Bernstein operators: Theory and applications. Iranian Journal of Science and Technology, Transactions A: Science. 44, 1119-1124 (2020).
  • [17] Alemdar, M. E., Duman, O.: General summability methods in the approximation by Bernstein–Chlodovsky operators. Numerical Functional Analysis and Optimization. 42(5), 497-509 (2021).

Korovkin Type Theorem for Modified Bernstein Operators via A-Statistical Convergence and Power Summability Method

Year 2025, Volume: 13 Issue: 2, 84 - 91, 26.06.2025

Tuğba Değirmenci , Merve Ilkhan Kara

https://doi.org/10.36753/mathenot.1657527

Abstract

In this study, we investigate the approximation properties of modified Bernstein operators through the lens of A-statistical convergence and power summability methods. Our main objective is to establish a Korovkin-type approximation theorem in this generalized setting. By incorporating statistical convergence, we aim to provide broader and more powerful approximation results that can be applied in various contexts where classical convergence criteria may fail or be insufficient.

Keywords

A-statistical convergence Bernstein operators Korovkin type theorem Power summability method

References

  • [1] Fast, H.: Sur la convergence statistique. Colloquium Mathematicum. 2, 241–244 (1951).
  • [2] Fridy, J. A., Miller, H. I.: A matrix characterization of statistical convertence. Analysis 11. 11(1), 59-66 (1991).
  • [3] Braha, N. L.: Some properties of Baskakov–Schurer-Szász operators via power summability method. Quaestiones Mathematicae. 42(10), 1411-1426 (2019).
  • [4] Kratz, W., Stadtmuller, U.: Tauberian theorems for Jp-summability. Journal of Mathematical Analysis and Applications 139, 362–371 (1989).
  • [5] Stadtmuller, U., Tali, A.: On certain families of generalized Nörlund methods and power series methods. Journal of Mathematical Analysis and Applications 238(1), 44–66 (1999).
  • [6] Başar, F.: Summability theory and its applications. Bentham Science Publishers. (2012).
  • [7] Braha, N. L.: Some weighted equi-statistical convergence and Korovkin type-theorem. Results in Mathematics. 70(3–4), 433–446 (2016).
  • [8] Taş, E.: Some results concerning Mastroianni operators by power series method. Communications Faculty of Sciences University of Ankara Series A1 Mathematics and Statistics. 63(1), 187-195 (2016).
  • [9] Taş, E., Yurdakadim, T.: Approximation by positive linear operators in modular spaces by power series method. Positivity. 21(4), 1293-1306 (2017).
  • [10] Boos, J.: Classical and modern methods in summability. Oxford University Press. Oxford (2000).
  • [11] Campiti, M., Metafune, G.: L Lp-convergence of Bernstein–Kantorovich-type operators. Annales Polonici Mathematici. 63(3), 273-280 (1996).
  • [12] Braha, N. L., Loku, V., Srivastava, H. M.: Λ2-Weighted statistical convergence and Korovkin and Voronovskaya type theorems. Applied Mathematics and Computation. 266, 675-686 (2015).
  • [13] Braha, N.L.: Some properties of new modified Szász–Mirakyan operators in polynomial weight spaces via power summability method. Bulletin of Mathematical Analysis and Applications. 10(3), 53-65 (2018).
  • [14] Uğur, K., Braha, N.L., Srivastava, H. M.: Statistical weighted B-summability and its applications to approximation theorems. Applied Mathematics and Computation. 302, 80–96 (2017).
  • [15] Loku, V., Braha, N. L.: Some weighted statistical convergence and Korovkin type-theorem. Journal of Inequalities and Special Functions. 8(3), 139–150 (2017).
  • [16] Usta, F: On new modification of Bernstein operators: Theory and applications. Iranian Journal of Science and Technology, Transactions A: Science. 44, 1119-1124 (2020).
  • [17] Alemdar, M. E., Duman, O.: General summability methods in the approximation by Bernstein–Chlodovsky operators. Numerical Functional Analysis and Optimization. 42(5), 497-509 (2021).
There are 17 citations in total.

Details

Primary Language English
Subjects Approximation Theory and Asymptotic Methods
Journal Section Articles
Authors

Tuğba Değirmenci 0000-0003-2826-9741

Merve Ilkhan Kara 0000-0002-0831-1474

Early Pub Date June 11, 2025
Publication Date June 26, 2025
Submission Date March 13, 2025
Acceptance Date June 10, 2025
Published in Issue Year 2025 Volume: 13 Issue: 2

Cite

APA Değirmenci, T., & Ilkhan Kara, M. (2025). Korovkin Type Theorem for Modified Bernstein Operators via A-Statistical Convergence and Power Summability Method. Mathematical Sciences and Applications E-Notes, 13(2), 84-91. https://doi.org/10.36753/mathenot.1657527
AMA Değirmenci T, Ilkhan Kara M. Korovkin Type Theorem for Modified Bernstein Operators via A-Statistical Convergence and Power Summability Method. Math. Sci. Appl. E-Notes. June 2025;13(2):84-91. doi:10.36753/mathenot.1657527
Chicago Değirmenci, Tuğba, and Merve Ilkhan Kara. “Korovkin Type Theorem for Modified Bernstein Operators via A-Statistical Convergence and Power Summability Method”. Mathematical Sciences and Applications E-Notes 13, no. 2 (June 2025): 84-91. https://doi.org/10.36753/mathenot.1657527.
EndNote Değirmenci T, Ilkhan Kara M (June 1, 2025) Korovkin Type Theorem for Modified Bernstein Operators via A-Statistical Convergence and Power Summability Method. Mathematical Sciences and Applications E-Notes 13 2 84–91.
IEEE T. Değirmenci and M. Ilkhan Kara, “Korovkin Type Theorem for Modified Bernstein Operators via A-Statistical Convergence and Power Summability Method”, Math. Sci. Appl. E-Notes, vol. 13, no. 2, pp. 84–91, 2025, doi: 10.36753/mathenot.1657527.
ISNAD Değirmenci, Tuğba - Ilkhan Kara, Merve. “Korovkin Type Theorem for Modified Bernstein Operators via A-Statistical Convergence and Power Summability Method”. Mathematical Sciences and Applications E-Notes 13/2 (June 2025), 84-91. https://doi.org/10.36753/mathenot.1657527.
JAMA Değirmenci T, Ilkhan Kara M. Korovkin Type Theorem for Modified Bernstein Operators via A-Statistical Convergence and Power Summability Method. Math. Sci. Appl. E-Notes. 2025;13:84–91.
MLA Değirmenci, Tuğba and Merve Ilkhan Kara. “Korovkin Type Theorem for Modified Bernstein Operators via A-Statistical Convergence and Power Summability Method”. Mathematical Sciences and Applications E-Notes, vol. 13, no. 2, 2025, pp. 84-91, doi:10.36753/mathenot.1657527.
Vancouver Değirmenci T, Ilkhan Kara M. Korovkin Type Theorem for Modified Bernstein Operators via A-Statistical Convergence and Power Summability Method. Math. Sci. Appl. E-Notes. 2025;13(2):84-91.
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