Periodic Korovkin Theorem via $P_{p}^{2}$-Statistical $\mathcal{A}$-Summation Process

Year 2022, , 156 - 162, 29.12.2022

Sevda Yıldız , Fadime Dirik , Kamil Demirci

https://doi.org/10.32323/ujma.1205420

Abstract

In the current research, we investigate and establish Korovkin-type approximation theorems for linear operators defined on the space of all $% 2\pi $-periodic and real valued continuous functions on $% %TCIMACRO{\U{211d} }% %BeginExpansion \mathbb{R} %EndExpansion ^{2}$ by means of $\mathcal{A}$-summation process via statistical convergence with respect to power series method. We demonstrate with an example how our theory is more strong than previously studied. Additionally, we research the rate of convergence of positive linear operators defined on this space.

Keywords

Korovkin theorem, periodic functions, power series method, rates of convergence, statistical convergence

References

• [1] A. Pringsheim, Zur theorie der zweifach unendlichen zahlenfolgen, Math. Ann., 53 (1900), 289-321.
• [2] F. Moricz, Statistical convergence of multiple sequences, Arch. Math. (Basel), 81 (2004), 82-89.
• [3] S. Baron, U. Stadtm¨uller, Tauberian theorems for power series methods applied to double sequences, J. Math. Anal. Appl., 211(2) (1997), 574-589.
• [4] M. U¨ nver, 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.
• [5] S. Yıldız, K. Demirci, F. Dirik, Korovkin theory via Pp-statistical relative modular convergence for double sequences, Rend. Circ. Mat. Palermo, II. Ser, (2022), 1-17.
• [6] R. F. Patterson, E. Savas¸, Uniformly summable double sequences, Studia Scientiarum Mathematicarum Hungarica, 44 (2007), 147-158.
• [7] E. Savas¸, B. E. Rhoades, Double summability factor theorems and applications, Math. Inequal. Appl., 10 (2007), 125-149.
• [8] P. P. Korovkin, On convergence of linear positive operators in the space of continuous functions, Doklady Akademii Nauk, 90 (1953), 961-964 (Russian).
• [9] O¨ .G. Atlıhan, C. Orhan, Summation process of positive linear operators, Computers and Mathematics with Applications, 56 (2008), 1188-1195.
• [10] N. S. Bayram, Strong summation process in locally integrable function spaces, Hacettepe Journal of Mathematics and Statistics, 45(3) (2016), 683-694.
• [11] N. S. Bayram, C. Orhan, A-Summation process in the space of locally integrable functions, Stud. Univ. Babes-Bolyai Math., 65 (2020), 255-268.
• [12] S. C¸ ınar, S. Yıldız, P-statistical summation process of sequences of convolution operators, Indian Journal of Pure and Applied Mathematics, 53(3) (2022), 648-659.
• [13] F. Dirik, K. Demirci, Korovkin type approximation theorem for functions of two variables in statistical sense, Turkish Journal of Mathematics, 34(1) (2010), 73-84.
• [14] F. Dirik, K. Demirci, B-statistical approximation for periodic functions, Studia Scientiarum Mathematicarum Hungarica, 47(3) (2010), 321-332.
• [15] K. Demirci, F. Dirik, Approximation for periodic functions via statistical s-convergence, Mathematical Communications, 16(1) (2011), 77-84.
• [16] K. Demirci, S. Orhan, Statistical relative approximation on modular spaces, Results in Mathematics, 71(3) (2017), 1167-1184.
• [17] K. Demirci, S. Yıldız, F. Dirik, Approximation via power series method in two-dimensional weighted spaces, Bulletin of the Malaysian Mathematical Sciences Society, 43(6) (2020), 3871-3883.
• [18] O. Duman, Statistical approximation for periodic functions, Demonstratio Mathematica, 36(4) (2003), 873-878.
• [19] A. D. Gadjiev, C. Orhan, Some approximation theorems via statistical convergence, The Rocky Mountain Journal of Mathematics, (2002), 129-138.
• [20] S. Orhan, K. Demirci, Statistical A-summation process and Korovkin type approximation theorem on modular spaces, Positivity, 18(4) (2014), 669-686.
• [21] V. I. Volkov, On the convergence of sequences of positive linear operators in the space of two variables, Dokl. Akad. Nauk. SSSR (N.S.) 115 (1957), 17-19.
• [22] O. Duman, E. Erkus¸, Approximation of continuous periodic functions via statistical convergence, Comput. Math. Appl., 52 (2006) 967-974.
• [23] K. Demirci, F. Dirik, Four-dimensional matrix transformation and rate of A-statistical convergence of periodic functions, Mathematical and Computer Modelling, 52 (2010), 1858-1866.