Yarı Tamlık, Yarı Düzgün Yakınsaklık ve Korovkin Tipi Teoremler Üzerine

Year 2024, Volume: 12 Issue: 2, 100 - 109, 29.08.2024

Alper Erdem , Tuncay Tunç

https://doi.org/10.20290/estubtdb.1410365

Abstract

Bu çalışmada neredeyse düzgün yakınsaklık, yarı düzgün yakınsaklık ve yarı tamlık kavramı gibi çeşitli yakınsaklık biçimlerine dayanan Korovkin tipi teoremler incelenmiştir. Yukarıda bahsedilen yakınsama türlerinin noktasal ve düzgün yakınsaklık arasında olduğu bilindiğinden, Korovkin teoreminde bu koşulların
hafifletilebileceği fark edilmiştir.

Keywords

Function Sequences, Convergence, Exhaustiveness, Korovkin Theorem

ON SEMI-EXHAUSTIVENESS, SEMI-UNIFORM CONVERGENCE AND KOROVKIN-TYPE THEOREMS

Abstract

In this study, we scrutinize the Korovkin-type theorems based on various forms of convergence, such as almost uniform convergence, semi-uniform convergence, and the concept of semi-exhaustiveness. Since it is known that the convergence types mentioned above are between point-wise and uniform convergence, it will be noticed that the circumstances can be mitigated in the Korovkin theorem.

Keywords

Function Sequences, Convergence, Exhaustiveness, Korovkin Theorem

References

• [1] Albayrak H, Pehlivan S. Filter exhaustiveness and F−α-convergence of function sequences. Filomat, 2013; 27 (8), 1373−1383.
• [2] Altomare F. Korovkin-type theorems and local approximation problems. Expositiones Mathematicae, 2022; 40 (4), 1229−1243.
• [3] Anastassiou GA, Duman O. Towards intelligent modeling: Statistical approximation theory. Springer, Berlin, 2011.
• [4] Athanassiadou E, Boccuto A, Dimitriou X, Papanastassiou N. Ascoli-type theorems and ideal (α)-convergence. Filomat, 2012; 26 (2), 397−405.
• [5] Bardaro C, Boccuto A, Demirci K, Mantellini I, Orhan S. Triangular A-Statistical approximation by double sequences of positive linear operators. Results in Mathematics, 2015; 68, 271–291.
• [6] Boccuto A, Demirci K and Yildiz S. Abstract korovkin-type theorems in the filter setting with respect to relative uniform convergence. Turkish J. of Mathematics, 2020; 44 (4), 1238–1249.
• [7] Caserta A, Kočinac LD. On statistical exhaustiveness. Applied Mathematics Letters, 2012; 25 (10), 1447–1451.
• [8] Das S, Ghosh A. A study on statistical versions of convergence of sequences of functions. Mathematica Slovaca, 2022; 72 (2), 443–458.
• [9] Demirci K, Boccuto A, Yıldız S, Dirik F. Relative uniform convergence of a sequence of functions at a point and korovkin-type approximation theorems. Positivity, 2020; 24, 1–11.
• [10] Demirci K, Orhan S. Statistically relatively uniform convergence of positive linear operators. Results in Mathematics, 2016; 69, 359–367.
• [11] Drozdowski R, Jedrzejewski J, Sochaczewska A. On the almost uniform convergence. Scientific Issues of Jan Dlugosz University in Czestochowa Mathematics, 2013; 18, 11–17.
• [12] Duman O, Özarslan MA, Erkuş-Duman E. Rates of ideal convergence for approximation operators. Mediterranean. Journal of Mathematics, 2010; 7, 111–121.
• [13] Ewert J. Almost uniform convergence. Periodica Mathematica Hungarica, 1993; 26 (1), 77–84.
• [14] Gadjiev AD, Orhan C. Some approximation theorems via statistical convergence. The Rocky Mountain Journal of Mathematics, 2002; 32, 129–138.
• [15] Ghosh A. I*−α convergence and I*−exhaustiveness of sequences of metric functions. Matematicki Vesnik, 2022; 74 (2), 110–118.
• [16] Gregoriades V, Papanastassiou N. The notion of exhaustiveness and Ascoli-type theorems. Topology and its Applications, 2008; 155 (10), 1111–1128.
• [17] Karakuş S, Demirci K, Duman O. Statistical approximation by positive linear operators on modular spaces. Positivity, 2010; 14, 321–334.
• [18] Korovkin PP. Linear operators and approximation theory, Hindustan Publising Corp., 1960.
• [19] Kosar C, Kosar NP. Simultaneous statistical approximation of analytic functions in annulus by k-positive linear operators. Journal of Mathematical Analysis, 2019; 10 (2), 46–57.
• [20] Papanastassiou N. A note on convergence of sequences of functions. Topology and its Applications, 2020; 275, 107017.
• [21] Toyganözü ZH, Pehlivan S. Some results on exhaustiveness in asymmetric metric spaces. Filomat, 2015; 29 (1), 183–192.
• [22] Tunc T, Erdem A. Korovkin-type theorems via some modes of convergence. Filomat, 2024; 38 (2), 523−530.
• [23] Zeren Y, Ismailov M, Karacam C. Korovkin-type theorems and their statistical versions in grand Lebesgue spaces. Turkish Journal of Mathematics, 2020; 44 (3), 1027–1041.