Approximation theorems using the method of $\mathcal{I}_{2}$-statistical convergence

Year 2024, Volume: 5 Issue: 2, 79 - 87, 31.12.2024

Fadime Dirik , Kamil Demirci , Sevda Yıldız

https://doi.org/10.54559/jauist.1581390

Abstract

In this study, we utilize the concept of $\mathcal{I}$-statistical convergence for double sequences to establish a general approximation theorem of Korovkin-type for double sequences of positive linear operators $(PLOs)$ mapping from $H_{\omega }\left( X\right) $ to $C_{B}\left( X\right) $ where $% X=\left[ 0,\infty \right) \times \left[ 0,\infty \right) .$ We then present an example that demonstrates the applicability of our new main result in cases where classical and statistical approaches are not sufficient. Furthermore, we compute the convergence rate of these double sequences of positive linear operators by employing the modulus of smoothness.

Keywords

Double sequence, statistical convergence, $\mathcal{I}_{2}$-statistical convergence, Korovkin theorem, the Bleimann Butzer and Hahn operator

Ethical Statement

No approval from the Board of Ethics is required.

References

• E. Altiparmak, Ö. G. Atlihan, A Korovkin-type approximation theorem for positive linear operators in Hω(K) via power series method, Sarajevo Journal of Mathematics 19 (2) (2023) 183–191.
• C. Belen, M. Yildirim, On generalized statistical convergence of double sequences via ideals, Annali Dell’universita’Di Ferrara 58, (2012) 11–20.
• K. Demirci, F. Dirik, A Korovkin type approximation theorem for double sequences of positive linear operators of two variables in A-statistical sense, Bulletin of the Korean Mathematical Society 47 (4) (2010) 825–837.
• S. Dutta, R. Ghosh, Korovkin type approximation theorem on an infinite interval in A𝓧 -statistical sense, Acta Mathematica Universitatis Comenianae 89 (1) (2019) 131–142.
• E. Erku, O. Duman, A-statistical extension of the Korovkin type approximation theorem, Proceedings of the Indian Academy of Sciences-Mathematical Sciences 115 (4) (2005) 499–508.
• A. Esi, M. K. Ozdemir, N. Subramanian, Korovkin-type approximation theorem for Bernstein Stancu operator of rough statistical convergence of triple sequence, Boletim da Sociedade Paranaense de Matematica 38 (7) (2020) 69–83.
• A. Gadjiev, Ö. Çakar, On uniform approximation by Bleimann, Butzer and Hahn operators on all positive semi-axis, Transactions of National Academy of Sciences of Azerbaijan. Series of Physical-Technical and Mathematical Sciences 19 (5) (1999) 21–26.
• M. Mursaleen, A. Alotaibi, Korovkin type approximation theorem for statistical A-summability of double sequences, Journal of Computational Analysis and Applications 15 (6) (2013) 1036–1045.
• S. Konca, Weighted lacunary 𝓧-statistical convergence, Journal of the Institute of Science and Technology 7 (1) (2017) 267–277.
• S. Orhan, F. Dirik, K. Demirci, A Korovkin type approximation theorem for double sequences defined on Hω(X) in statistical A-summability sense, Miskolc Mathematical Notes 15 (2) (2014) 625–633.
• M. A. Ozarslan, New Korovkin type theorem for non-tensor M eyer-Konig and Zeller operators, Results in Mathematics 69 (2016) 327–343.
• M. Ünver, C. Orhan, Statistical convergence with respect to power series methods and applications to approximation theory, Numerical Functional Analysis and Optimization 40 (5) (2019) 535–547.
• S. Yıldız, K. Demirci, F. Dirik, Korovkin theory via Pp-statistical relative modular convergence for double sequences, Rendiconti del Circolo Matematico di Palermo Series 2 72 (2) (2023) 1125–1141.
• A. Pringsheim, Zur theorie der zweifach unendlichen zahlenfolgen, Mathematische Annalen 53 (3) (1900) 289–321.
• F. Moricz, Statistical convergence of multiple sequences, Archiv der Mathematik 81 (2003) 82–89.
• P. Kostyrko, W. Wilczynski, T. Salat, 𝓧-convergence, Real Analysis Exchange 26 (2000) 669–686.
• P. Das, P. Kostyrko, W. Wilczyn´ski, P. Malik, 𝓧 and 𝓧∗-convergence of double sequences, Mathematica Slovaca 58 (5) (2008) 605–620.
• E. Savas, P. Das, A generalized statistical convergence via ideals, Applied mathematics letters 24 (6) (2011) 826–830.
• G. G. Lorentz, Approximation of functions, Holt-Rinehart-Wilson, New York, 1966.
• G. A. Anastassiou, S. G. Gal, Approximation theory: moduli of continuity and global smoothness preservation, Springer Science & Business Media, New York, 2012.