Approximation by statistical convergence with respect to power series methods

Year 2022, Volume: 51 Issue: 4, 1108 - 1120, 01.08.2022

Nilay Şahin Bayram , Sevda Yıldız

https://doi.org/10.15672/hujms.1022072

Cited By: 4

Abstract

In the present work, using statistical convergence with respect to power series methods, we obtain various Korovkin-type approximation theorems for linear operators defined on derivatives of functions. Then we give an example satisfying our approximation theorem. We study certain rate of convergence related to this method. In the final section we summarize these results to emphasize the importance of the study.

Keywords

Korovkin type approximation, statistical convergence with respect to power series method, rate of convergence, linear operators

References

• [1] F. Altomare and M. Campiti, Korovkin Type Approximation Theory and Its Applications, de Gruyter, Berlin, 1994. [2] G.A. Anastassiou and O. Duman, On relaxing the positivity condition of linear operators in statistical Korovkin-type approximations J. Comput. Anal. Appl. 11 (1), 7-19, 2009.
• [3] G.A. Anastassiou and O. Duman, Towards Intelligent Modeling: Statistical Approximation Theory, Intelligent Systems Reference Library, 14, 117-129, 2016.
• [4] C. Bardaro, A. Boccuto, K. Demirci, I. Mantellini and S. Orhan, Triangular A-statistical approximation by double sequences of positive linear operators, Results Math. 68 (3), 271-291, 2015.
• [5] C. Bardaro, A. Boccuto, K. Demirci, I. Mantellini and S. Orhan, Korovkin-Type Theorems for Modular $\varphi$-A-Statistical Convergence Journal of Function Spaces, 2015.
• [6] C. Belen, M. Yıldırım and C. Sümbül, On statistical and strong convergence with respect to a modulus function and a power series method, Filomat, 34 (12), 3981- 3993, 2020.
• [7] N.L. Braha, T. Mansour, M. Mursaleen and T. Acar, Convergence of $\lambda$-Bernstein operators via power series summability method, J. Appl. Math. Comput. 65 (1-2), 125-146, 2021.
• [8] S. Çınar and S. Yıldız, P−statistical summation process of sequences of convolution operators, Indian J Pure Appl Math., https://doi.org/10.1007/s13226-021-00156-y, 2021.
• [9] F.J.M. Delgado, V.R. Gonzáles and D. C. Morales, Qualitative Korovkin type results on conservative approximation, J. Approx. Theory, 94, 144-159, 1998.
• [10] O. Duman and C. Orhan, Statistical approximation by positive linear operators, Studia Math. 161, 187-197, 2004.
• [11] O. Duman and C. Orhan, Statistical approximation in the space of locally integrable functions, Publ. Math. Debrecen, 63, 133-144, 2003.
• [12] H. Fast, Sur la convergence statistique, Colloq. Math. 2, 241-244, 1951.
• [13] A.D. Gadjiev and C. Orhan, Some approximation theorems via statistical convergence, Rocky Mountain J. Math. 32, 129-137, 2002.
• [14] P.P. Korovkin, Linear Operators and The Theory of Approximation, India, Delhi 1960.
• [15] W. Kratz and U. Stadtmüller, Tauberian theorems for $J_{p}$−summability, J. Math. Anal. Appl. 139, 362-371, 1989.
• [16] V. Loku, N.L. Braha, T. Mansour and M. Mursaleen, Approximation by a power series summability method of Kantorovich type Szász operators including Sheffer polynomials, Adv. Difference Equ. 165, 1-13, 2021.
• [17] S. Orhan and K. Demirci, Statistical A−summation process and Korovkin type approximation theorem on modular spaces, Positivity, 18 (4), 669-686, 2014.
• [18] S. Orhan and K. Demirci, Statistical approximation by double sequences of positive linear operators on modular spaces, Positivity 19 (1), 23-36, 2015.
• [19] I. Özgüç and E. Tas, A Korovkin-type approximation theorem and power series method, Results Math. 69, 497-504, 2016.
• [20] J.K. Singh, P.N. Agrawal and A. Kajla, Approximation by modified q-Gamma type op- erators via A-statistical convergence and power series method, Linear and Multilinear Algebra, DOI: 10.1080/03081087.2021.1960260, 2021.
• [21] D. Söylemez and M. Ünver,Rates of Power Series Statistical Convergence of Positive Linear Operators and Power Series Statistical Convergence of q-Meyer–König and Zeller Operators, Lobachevskii Journal of Mathematics 42 (2), 426-434, 2021.
• [22] U. Stadtmüller and A. Tali, On certain families of generalized Nörlund methods and power series methods, J. Math. Anal. Appl. 238, 44-66, 1999.
• [23] H. Steinhaus, Sur la convergence ordinaire et la convergence asymptotique, Col loq. Math. 2, 73-74 (1951).
• [24] N. Şahin Bayram, Criteria for statistical convergence with respect to power series methods, Positivity 25 (3), 1097-1105, 2021.
• [25] P. Şahin Okçu and F. Dirik, A Korovkin-type theorem for double sequences of positive linear operators via power series method, Positivity 22, 209-218, 2018.
• [26] E. Tas, T. Yurdakadim and Ö.G. Atlıhan, Korovkin type approximation theorems in weighted spaces via power series method, Oper. Matrices 12 (2), 529-535, 2018.
• [27] E. Tas and Ö.G. Atlıhan,Korovkin type approximation theorems via power series method, São Paulo J. Math. Sci. 13, 696-707, 2019.
• [28] M. Ünver, Abel transforms of positive linear operators, AIP Conf. Proc. 1558 (1), 1148-1151, 2013.
• [29] M. Ünver,Abel transforms of positive linear operators on weighted spaces, Bull. Belg. Math. Soc. Simon Stevin 21 (5), 813-822, 2014.
• [30] M. Ünver and C. Orhan, Statistical convergence with respect to power series methods and applications to approximation theory, Numerical and Functional Analysis and Optimization 40 (5), 533-547, 2019.