A General Korovkin Result Under Generalized Convergence

Year 2019, , 81 - 88, 01.06.2019

Pedro Garrancho

https://doi.org/10.33205/cma.530987

Cited By: 4

Abstract

In this paper the classic result of Korovkin about the convergence of sequences of functions defined from sequences of linear operators is reformulated in terms of generalized convergence. This convergence extends some others given in the literature. The operator of the sequence fulfill a shape preserving property more general than the positivity. This property is related with certain extension of the notion of derivative. This extended derivative is precisely the object of the approximation process. The study is completed by analysing the conditions for the existence of an asymptotic formula, from which some interesting consequences are derived as a local version of the shape preserving property. Finally, as applications of the previous results, the author use the following notion of generalized convergence, an extension of Nörlund-Cesaro summability given by V. Loku and N. L. Braha in 2017. A way to transfer a notion of generalized convergence to approximation theory by means of linear operators is showed.

Keywords

Korovkin results, asymptotic condition, generalized convergence.

References

• [1] T. Acar and S. A. Mohiuddine, Statistical (C; 1)(E; 1) Summability and Korovkin’s Theorem, Filomat, 30:02, 2016, 387–393.
• [2] D. Cárdenas-Morales, P. Garrancho and I. Ra¸sa, Bernstein-type operators which preserve polynomials, Comput. Math. Appl., 62, 2011, 158–163.
• [3] D. Cárdenas-Morales and P. Garrancho, B􀀀statistical A-summability in conservative approximation, Math. Inequal. Appl.,19(3), 2016, 923–936.
• [4] O. Duman, M. K. Khan and C. Orhan, A-Statistical convergence of approximating operators, Math. Inequal. Appl. 6, 2003, 689–699.
• [5] H. Fast, Sur la convergence statistique, Colloq. Math., 2, 1951, 241–244.
• [6] A. D. Gadjiev and C. Orhan, Some approximation theorems via statistical convergence, Rocky Mountain J. Math., 32, 2002, 129–138.
• [7] P. Garrancho and D. Cárdenas-Morales, A converse of asymptotic formulae in simultaneous approximation, Appl. Math. and Comp., 217, 2010, 2676–2683.
• [8] A. Karaisa, Statistical $\alpha\beta$-Summability and Korovkin Type Approximation Theorem, Filomat ,30:13, 2016, 3483–3491.
• [9] V. Karakaya and A. Karaisa, Korovkin type approximation theorems for weighted $\alpha\beta$-statistical convergence, Bull. Math. Sci., 5, 2015, 159–169.
• [10] J. P. King and J. J. Swetits, Positive linear operators and summability, Austral J. Math., 11, 1970, 281–291.
• [11] P. P. Korovkin, Linear operators and approximation theory, Hindustan Publishing Corp., Delhi, India, 1960.
• [12] V. Loku and N. L. Braha, Tauberian Theorems by Weighted Summability Method, Armenian J. of Math., 9(1), 2017, 35–42.
• [13] A.-J. López-Moreno and F.-J. Muñoz-Delgado, Asymptotic expansion of multivariate conservative linear operators, J. Comput. Appl. Math, 150, 2003, 219–251.
• [14] G. G. Lorentz, A contribution to the theory of divergent sequences, Acta. Math., 80, 1948, 167–190.
• [15] H. Sharma, R. Maurya and C. Gupta, Approximation properties of Kantorovich Type Modifications of p; q-Meyer-König- Zeller Operators, Constr. Math. Anal., 1(1), 2018, 58–72.