Theorems of Second Korovkin Type with respect to Triangular $A$-Statistical Convergence

Year 2023, Volume: 6 Issue: 1, 15 - 22, 28.03.2023

Selin Çınar

https://doi.org/10.32323/ujma.1207010

Abstract

This article is a continuation of our previous works. We mainly investigate a Korovkin type theorem for double sequences of positive linear operators defined in the space of all $2\pi $-periodic and real valued continuous functions on the real two-dimensional space with help of the concept of triangular $A$-statistical convergence, which is a kind of statistical convergence for double real sequences. Also, we analyze the rate of convergence of double operators in this via modulus of continuity.

Keywords

Positive linear operator, Korovkin type theorem, triangular A-statistical convergence

References

• [1] H. Fast, Sur la convergence statistique, Colloq. Math. 2 (1951), 241-244.
• [2] H. Steinhaus, Sur la convergence ordinaire et la convergence asymtotique, Colloq. Math. 2 (1951), 73-74.
• [3] C. Bardaro, A. Boccuto, K. Demirci, I. Mantellini, S. Orhan, Triangular A-statistical approximation by double sequences of positive linear operators, Results in Mathematics, 68 (2015), 271-291.
• [4] C. Bardaro, A. Boccuto, K. Demirci, I. Mantellini, S. Orhan, Korovkin-Type Theorems for Modular Y-A-Statistical Convergence, Journal of Function Spaces, 2015 (2015), 1-11.
• [5] K. Demirci, F. Dirik, P. Okc¸u, Approximation in Triangular Statistical Sense to B-Continuous Functions by Positive Linear Operators, Annals of the Alexandru Ioan Cuza University-Mathematics, 63(3) (2017).
• [6] S. C¸ ınar, Triangular A-statistical relative uniform convergence for double sequences of positive linear operator, Facta Universitatis. Series: Mathematics and Informatics, (2021) 065-077.
• [7] S. C¸ ınar, S. Yıldız, K. Demirci, Korovkin type approximation via triangular A-statistical convergence on an infinite interval, Turkish Journal of Mathematics 45(2) (2021), 929-942.
• [8] P. P. Korovkin, Linear Operators and Approximation Theory, Hindustan Publ. Co., Delhi, 1960.
• [9] C. Bardaro, I. Mantellini, Korovkin’s theorem in modular spaces, Commentationes Math. 47 (2007), 239-253.
• [10] K. Demirci, A. Boccuto, S. Yıldız, F. Dirik, Relative uniform convergence of a sequence of functions at a point and Korovkin-type approximation theorems, Positivity, 24(1) (2020), 1-11.
• [11] K. Demirci, S. Orhan, Statistically relatively uniform convergence of positive linear operators, Results Math., 69 (2016), 359-367.
• [12] K. Demirci, S. Orhan, B. Kolay, Relative Hemen Hemen Yakınsaklık ve Yaklas¸ım Teoremleri, Sinop U¨ niversitesi Fen Bilimleri Dergisi, 1(2) (2016), 114-122.
• [13] 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.
• [14] K. Demirci, F. Dirik, Approximation for periodic functions via statistical s-convergence, Mathematical Communications, 16(1) (2011), 77-84.
• [15] K. Demirci, F. Dirik, S. Yıldız, Approximation via equi-statistical convergence in the sense of power series method, Revista de la Real Academia de Ciencias Exactas, 116(2) (2022), 1-13.
• [16] O. Duman, Statistical approximation for periodic functions, Demons. Math., 36(4) (2003), 873-878. [17] O. Duman, E. Erkus¸, Approximation of continuous periodic functions via statistical convergence, Comput. Math. Appl., 52 (2006) 967-974.
• [18] O. Duman, M. K. Khan, C. Orhan, A-statistical convergence of approximating operators, Math. Inequal. Appl., 6 (2003) 689-699.
• [19] A. D. Gadjiev, C. Orhan, Some approximation theorems via statistical convergence, Rocky Mountain J. Math., 32 (2002), 129-138.
• [20] 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.
• [21] A. Pringsheim, Zur theorie der zweifach unendlichen zahlenfolgen, Math. Ann., 53 (1900), 289-321.
• [22] H.J. Hamilton, Transformations of multiple sequences, Duke Math. J., 2 (1936), 29-60.
• [23] G.M. Robison, Divergent double sequences and series, Amer. Math. Soc. Transl., 28 (1926), 50-73.
• [24] F. Moricz, Statistical convergence of multiple sequences, Arch. Math. (Basel), 81 (2004), 82-89.
• [25] G.H. Hardy, Divergent Series, Oxford Univ. Press, London, 1949.
• [26] K. Demirci, F. Dirik, Four-dimensional matrix transformation and rate of A-statistical convergence of periodic functions, Math. Comput. Modelling, 52 (2010), 1858-1866.