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.