Antonıo BOCCUTO, Kamil DEMİRCİ, Sevda YILDIZ

Abstract Korovkin-type theorems in the filter setting with respect to relative uniform convergence

We prove a Korovkin-type approximation theorem using abstract relative uniform filter convergence of a net of functions with respect to another fixed filter, a particular case of which is that of all neighborhoods of a point, belonging to the domain of the involved functions. We give some examples, in which we show that our results are strict generalizations of the classical ones.

Keywords:

Korovkin theorem, relative uniform convergence, bivariate Kantorovich operator,

___

[1] Agrawal MR, Tewari UB. On existence of finite universal Korovkin in the centre of group algebras. Monatshefte für Mathematik 1997; 123: 1-20.
[2] Altomare F. Korovkin-type theorems and approximation by positive linear operators. Surveys in Approximation Theory 2010; 5: 92-164.
[3] Altomare F, Campiti M. Korovkin-type approximation theory and its applications. Berlin: De Gruyter Studies in Mathematics, 1994.
[4] Altomare F, Diomede S. Contractive Korovkin subsets in weighted spaces of continuous functions. Rendiconti del Circolo Matematico di Palermo 2001; 50 (3): 547-568.
[5] Anastassiou GA, Duman O. Towards Intelligent Modeling: Statistical Approximation Theory. Vol. 14. Berlin: Springer, 2011.
[6] Angeloni L, Vinti G. Rate of approximation for nonlinear integral operators with application to signal processing. Different Integral Equations 2005; 18: 855-890.
[7] Angeloni L, Vinti G. Convergence and rate of approximation for linear integral operators in BVφ -spaces in multidimensional setting. Journal of Mathematical Analysis and Applications 2009; 349 (2): 317-334.
[8] 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.
[9] Bardaro C, Boccuto A, Demirci K, Mantellini I, Orhan S. Korovkin-type theorems for modular Ψ-A-statistical convergence. Journal of Function Spaces 2015; 2015: 11 pages, Article ID 160401.
[10] Bardaro C, Boccuto A, Dimitriou X, Mantellini I. Abstract Korovkin-type theorems in modular spaces and applications. Central European Journal of Mathematics 2013; 11 (10): 1774-1784.
[11] Bardaro C, Karsli H, Vinti G. On pointwise convergence of linear integral operators with homogeneous kernels. Integral Transforms and Special Functions 2008; 19 (6): 429-439.
[12] Bardaro C and Mantellini I. A Korovkin theorem in multivariate modular function spaces. Journal of Function Spaces and Applications 2009; 7 (2): 105-120.
[13] Bardaro C, Musielak J, Vinti G. Nonlinear Integral Operators and Applications. Berlin: De Gruyter Studies in Mathematics, 2003.
[14] Boccuto A, Candeloro D, Sambucini AR. Vitali-type theorems for filter convergence related to vector lattice-valued modulars and applications to stochastic processes. Journal of Mathematical Analysis and Applications 2014; 419 (2): 818-838.
[15] Boccuto A, Candeloro D, Sambucini AR. L p spaces in vector lattices and applications. Mathematica Slovaca 2017; 67 (6): 1409-1426.
[16] Boccuto A, Dimitriou X. Rates of approximation for general sampling-type operators in the setting of filter convergence. Applied Mathematics and Computation 2014; 229: 214-226.
[17] Boccuto A, Dimitriou X. Korovkin-type theorems for abstract modular convergence. Results in Mathematics 2016; 60: 477-495.
[18] Bohman H. On approximation of continuous and of analytic functions. Arkiv för Matematik 1952; 2 (3): 43-56.
[19] Campiti M. Korovkin-type approximation in spaces of vector-valued and set-valued functions. Applicable Analysis 2019; 98 (13): 2486-2496.
[20] Chittenden EW. Relatively uniform convergence of sequences of functions. Transactions of the American Mathematical Society 1914; 15: 197-201.
[21] Chittenden EW. On the limit functions of sequences of continuous functions converging relatively uniformly. Transactions of the American Mathematical Society 1919; 20: 179-184.
[22] Chittenden EW. Relatively uniform convergence and classification of functions. Transactions of the American Mathematical Society 1922; 23: 1-15.
[23] Cluni F, Costarelli D, Minotti AM, Vinti G. Applications of sampling Kantorovich operators to thermographic images for seismic engineering. Journal of Computational Analysis and Applications 2015; 19: 602-617.
[24] Costarelli D, Sambucini AR. Approximation results in Orlicz spaces for sequences of Kantorovich max-product neural network operators. Results in Mathematics 2018; 73 (1): 1-15.
[25] Costarelli D, Vinti G. Approximation by nonlinear multivariate sampling Kantorovich type operators and applications to image processing. Numerical Functional Analysis and Optimization 2013; 34: 819-844.
[26] Costarelli D, Vinti G. Convergence for a family of neural network operators in Orlicz spaces. Mathematische Nachrichten 2017; 290: 226-235.
[27] Costarelli D, Vinti G. Approximation theorems for a family of multivariate neural network operators in Orlicz-type spaces. Ricerche di Matematica 2018; 67 (2): 387-399.
[28] Demirci K, Boccuto A, Yildiz S, Dirik F. Relative uniform convergence of a sequence of functions at a point and Korovkin-type approximation theorems. Positivity 2020; 24: 1-11.
[29] Demirci K, Orhan S. Statistically relatively uniform convergence of positive linear operators. Results in Mathematics 2016; 69: 359-367.
[30] Dolecki S, Mynard S. Convergence foundations of topology. Hackensack, NJ, USA: World Scientific Publishing Co., 2016.
[31] Donner K. Korovkin theorems in L p spaces. Journal of Functional Analysis 1981; 42 (1): 12-28.
[32] Duman O, Özarslan MA, Erkuş-Duman E. Rates of ideal convergence for approximation operators. Mediterranean Journal of Mathematics 2010; 7: 111-121.
[33] Gadjiev AD, Orhan C. Some approximation theorems via statistical convergence. The Rocky Mountain Journal of Mathematics 2002; 32: 129-138.
[34] Karakuş S, Demirci K, Duman O. Statistical approximation by positive linear operators on modular spaces. Positivity 2010; 14: 321-334.
[35] Katětov M. Products of filters. Commentationes Mathematicae Universitatis Carolinae 1968; 9: 173-189.
[36] Kitto W, Wulbert DE. Korovkin approximations in L p spaces. Pacific Journal of Mathematics 1976; 63: 153-167.
[37] Klippert J, Williams G. Uniform convergence of a sequence of functions at a point. International Journal of Mathematical Education in Science and Technology 2002; 33: 51-58.
[38] Korovkin PP. On convergence of linear positive operators in the spaces of continuous functions (Russian). Doklady Akademii Nauk SSSR 1953; 90: 961-964.
[39] Korovkin PP. Linear operators and approximation theory. New Delhi, India: Hindustan Publishing Company, 1960.
[40] Kostyrko P, Šalát T, Wilczyński W. I -convergence. Real Analysis Exchange 2001; 26 (2): 669-685.
[41] Maligranda L. Korovkin theorem in symmetric spaces. Commentationes Mathematicae Prace Matematyczne 1987; 27: 135-140.
[42] Niven I, Zuckerman HS. An Introduction to the Theory of Numbers. 4th ed. New York, NY, USA: Wiley, 1980.
[43] Pinkus A. Approximation theory of the MLP model in neural networks. Acta Numerica 1999; 8: 143-195.
[44] Pinkus A. Density in approximation theory. Surveys in Approximation Theory 2005; 1: 1-45.
[45] Pop OT, Farcas MD. About the bivariate operators of Kantorovich type. Acta Mathematica Universitatis Comenianae 2009; Vol. LXXVIII (1): 43-52.
[46] Renaud PF. A Korovkin theorem for abstract Lebesgue spaces. Journal of Approximation Theory 2000; 102: 13-20.
[47] Soardi PM. On quantitative Korovkin’s theorem in multivariate Orlicz spaces. Mathematica Japonica 1998; 48: 205-212.
[48] Vinti G. A general approximation result for nonlinear integral operators and applications to signal processing. Applicable Analysis 2001; 79: 217-238.
[49] Yilmaz B, Demirci K, Orhan S. Relative modular convergence of positive linear operators. Positivity 2016; 20: 565-577.