Finally, an estimate for the rate at which $\mathcal{I}$-equi-statistical convergence occurs is provided, thus quantifying the efficiency of approximation under this framework.
📌 3 Key Takeaways
🎯
Objective & Hypothesis:In the present work, a novel type of convergence is introduced by incorporating the notions of $\mathcal{I}$-statistical convergence and equi-statistical convergence.
📊
Key Findings & Data:Moreover, the paper provides a concrete example that illustrates the practical significance of the new convergence concept.
💡
Conclusion & Impact:Finally, an estimate for the rate at which $\mathcal{I}$-equi-statistical convergence occurs is provided, thus quantifying the efficiency of approximation under this framework.
In the present work, a novel type of convergence is introduced by incorporating the notions of $\mathcal{I}$-statistical convergence and equi-statistical convergence. These offer a more extensive framework in comparison to the classical concepts of $\mathcal{I}$-convergence and statistical convergence. Within this generalized setting, two fundamental approximation results are established and proven: one in the sense of Korovkin-type theorems and another in the sense of a Voronovskaya-type result. Moreover, the paper provides a concrete example that illustrates the practical significance of the new convergence concept. Finally, an estimate for the rate at which $\mathcal{I}$-equi-statistical convergence occurs is provided, thus quantifying the efficiency of approximation under this framework.
Dirik, F., Demirci, K., Yıldız, S. (2026). Approximation Theorems and $\mathcal{I}$-Equi Statistical Convergence. Communications Faculty of Sciences University of Ankara Series A1 Mathematics and Statistics, Volume: 75 Issue: 2. https://doi.org/10.31801/cfsuasmas.1699182
Acu, A. M., Aral, A., and Rasa, I. (2023). New properties of operators preserving exponentials. RACSAM, 117(1):1.
Altomare, F. and Campiti, M. (1994). Korovkin-Type Approximation Theory and Its Applications. de Gruyter Stud. Math. 17. Walter de Gruyter, Berlin.
Aslan, R. (2025). Some approximation properties of Riemann-Liouville type fractional Bernstein- Stancu-Kantorovich operators with order of $\alpha$. Iranian Journal of Science, 49(2):481–494.
Balcerzak, M., Dems, K., and Komisarski, A. (2007). Statistical convergence and ideal convergence for sequences of functions. Journal of Mathematical Analysis and Applications, 328:715–729.
Çınar, S. and Yıldız, S. (2022). $P$-statistical summation process of sequences of convolution operators. Indian Journal of Pure and Applied Mathematics, 53:648–659.
Çınar, S., Yıldız, S., and Demirci, K. (2021). Korovkin type approximation via triangular $A$-statistical convergence on an infinite interval. Turkish Journal of Mathematics, 45(2):929–942.
Connor, J. (1992). ${R}$-type summability methods, cauchy criteria, $P$-sets and statistical convergence. Proceedings of the American Mathematical Society, 115(2):319–327.
Das, P., Savas, E., and Ghosal, S. (2011). On generalizations of certain summability methods using ideals. Applied Mathematics Letters, 24(4):1509–1514.
Debnath, S. and Rakshit, D. (2018). On $\mathcal{I}$-statistical convergence. Iranian Journal of Mathematical Sciences and Informatics, 13(2):101–109.
Demirci, K., Dirik, F., and Yıldız, S. (2021). Deferred nörlund statistical relative uniform convergence and Korovkin-type approximation theorem. Communications Faculty of Sciences University of Ankara Series A1 Mathematics and Statistics, 70(1):279–289.
Demirci, K., Dirik, F., and Yıldız, S. (2022). Approximation via equi-statistical convergence in the sense of power series method. Revista de la Real Academia de Ciencias Exactas, Físicas y Naturales. Serie A: Matemáticas, 116(2):1–13.
Demirci, K., Dirik, F., and Yıldız, S. (2025). Approximation results via power series method for sequences of monotone and sublinear operators. Results in Mathematics, 80(48):1–19.
Dirik, F. and Demirci, K. (2010a). Approximation in statistical sense to $n$-variate $B$-continuous functions by positive linear operators. Mathematica Slovaca, 60(6):877–886.
Dirik, F. and Demirci, K. (2010b). Four dimensional matrix transformation and rate of $A$-statistical convergence of continuous functions. Computers & Mathematics with Applications, 59(8):2976–2981.
Fast, H. (1951). Sur la convergence statistique. Colloquium Mathematicum, 2:241–244.
Gadjiev, A. D. and Orhan, C. (2002). Some approximation theorems via statistical convergence. Rocky Mountain Journal of Mathematics, 32:129–138.
Gogola, J., Macaj, M., and Višňai, T. (2011). On $\mathcal{I}_c^{(q)}$ -convergence. Annales Mathematicae et Informaticae, 38:27–36.
Gonska, H., Pitul, P., and Rasa, I. (2009). General King-type operators. Results in Mathematics, 53:279–286.
Karakuş, S., Demirci, K., and Duman, O. (2008). Equi-statistical convergence of positive linear operators. Journal of Mathematical Analysis and Applications, 339:1065–1072.
Korokin, P. P. (1960). Linear Operators and Approximation Theory. Hindustan Publishing Company, Delhi.
Kostyrko, P., Macaj, M., Salat, T., and Sleziak, M. (2005). $\mathcal{I}$-convergence and extremal $\mathcal{I}$-limit points. Mathematica Slovaca, 55(4):443–464.
Kostyrko, P., Salat, T., and Wilczyński, W. (2000–2001). $\mathcal{I}$-convergence. Real Analysis Exchange, 26(2):669–686.
Orhan, C. and Şahin Bayram, N. (2020). $A$-summation process in the space of locally integrable functions. Studia Universitatis Babes, -Bolyai Mathematica, 65(2):255–268.
Özger, F., Aslan, R., and Ersoy, M. (2025). Some approximation results on a class of Szász-Mirakjan- Kantorovich operators including non-negative parameter $\alpha$. Numerical Functional Analysis and Optimization, 46(6):461–484.
Şahin Bayram, N. (2016). Strong summation process in locally integrable function spaces. Hacettepe Journal of Mathematics and Statistics, 45(3):683–694.
Şahin Bayram, N. and Yıldız, S. (2022). Approximation by statistical convergence with respect to power series methods. Hacettepe Journal of Mathematics and Statistics, 51(4):1108–1120.
Steinhaus, H. (1951). Sur la convergence ordinaire et la convergence asymptotique. Colloquium Mathematicum, 2:73–74.
Ünver, M. and Orhan, C. (2019). Statistical convergence with respect to power series methods and applications to approximation theory. Numerical Functional Analysis and Optimization, 40(5):535–547.
Yıldız, S., Demirci, K., and Dirik, F. (2025). Approximation results on an infinite interval based on power series statistical sense. Dolomites Research Notes on Approximation, 18(2):17–24.
Yıldız, S. and Şahin Bayram, N. (2023). Construction of bivariate modified Bernstein-Chlodowsky operators and approximation theorems. Dolomites Research Notes on Approximation, 16(2):64–71.