Statistical Korovkin-type theorem for monotone and sublinear operators
DOI:
https://doi.org/10.24193/subbmath.2023.2.14Keywords:
Korovkin-type theorems, monotone and sublinear operator, nonlinear Choquet integral, statistical convergence.Abstract
In this paper we generalize the result on statistical uniform convergence in the Korovkin theorem for positive and linear operators in C([a, b]), to the more general case of monotone and sublinear operators. Our result is illustrated by concrete examples.
Mathematics Subject Classification (2010): 41A35, 41A36, 41A63.
Received 18 April 2022; Accepted 05 February 2023.
References
Agratini, O., Statistical convergence applied to Korovkin-type approximation theory, WSEAS Transactions on Mathematics, 16(2017), 183-186.
Agratini, O., From uniform to statistical convergence of binomial-type operators, in Advances in Summability and Approximation Theory, Springer, (2018), 169-179.
Agratini, O., Approximation properties of a class of linear operators, Mathematical Methods in the Applied Sciences, 36(17)(2013), 2353-2358.
Agratini, O., A-Statistical convergence of a class of integral operators, Applied Mathematics and Information Sciences, 6(2012), no. 2, 325-328.
Akdäg, S., Statistical e-convergence of double sequences on probabilistic normed spaces, Stud. Univ. Babeș-Bolyai Math., 64(2019), no. 2, 529-536.
Altomare, F., Korovkin-type theorems and positive operators, Surveys in Approximation Theory, 6(2010), 92-164.
Altomare, F., Campiti, M., Korovkin-Type Approximation Theory and Its Applications, de Gruyter Studies in Mathematics, 17, Berlin 1994, reprinted 2011.
Cárdenas-Morales, D., Garancho, P., Optimal linear approximation under general statistical convergence, in Advances in Summability and Approximation Theory, Springer, 2018, 191-202.
Cerreia-Vioglio, S., Maccheroni, F., Marinacci M., Montrucchio, L., Signed integral representations of comonotonic additive functionals, J. Math. Anal. Appl., 385(2012), 895- 912.
Connor, J., The statistical and strong p-Cesaro convergence of sequences, Analysis, 8(1988), 47-63.
Dellacherie, C., Quelques Commentaires sur les Prolongements de Capacités, Séminaire Probabilités V, Strasbourg, Lecture Notes in Math., 191, Springer-Verlag, Berlin-New York, 1970.
Denneberg, D., Non-Additive Measure and Integral, Kluwer Academic Publisher, Dordrecht, 1994.
Dirik, F., Weighted statistical convergence of Bögel continuous functions by positive linear operator, in Advances in Summability and Approximation Theory, Springer, 2012, 181-189.
Duman, O., Khan, M.K., Orhan, C., A-Statistical convergence of approximating operators, Math. Ineq. Appl., 6(2003), no. 4, 689-699.
Duman, O., A-Statistical convergence of sequences of convolution operators, Taiwanese J. Math., 12(2008), no. 2, 523-536.
Erdös, P., Tenenbaüm, G., Sur les densités de certaines suites dentiers, Proc. London Math. Soc., 59(1989), no. 3, 417-438.
Fast, H., Sur la convergence statistique, Colloq. Math., 2(1951), 241-244.
Fridy, J.A., On statistical convergence, Analysis, 5(1985), 301-313.
Fridy, J.A., Khan, M.K., Tauberian theorems via statistical convergence, J. Math. Anal. Appl., 228(1998), 73-95.
Fridy, J.A., Orhan, C., Lacunary statistical summability, J. Math. Anal. Appl., 173(1993), 497-504.
Gadjev, A.D., The convergence problem for a sequence of positive linear operators on unbounded sets, and theorems analogous to that of P.P. Korovkin, Soviet Math. Dokl., 15(1974), 1433-1436.
Gadjev, A.D., Orhan, C., Some approximation theorems via statistical convergence, Rocky Mountain J. Math., 32(2002), no. 1, 129-138.
Gal, S.G., Approximation by polynomial possibilistic integral operators, Ann. Acad. Rom. Sci. Ser. Math. Appl., 12(2020), 132-141.
Gal, S.G., Niculescu, C.P., A nonlinear extension of Korovkin’s theorem, Mediterr. J. Math., 17(2020), Article no. 145.
Gal, S.G., Niculescu, C.P., A note on the Choquet type operators, Aequationes Math., 95(2021), 433-447.
Gal, S.G., Niculescu, C.P., Choquet operators associated to vector capacities, J. Math. Anal. Appl., 479(2019), 903-925.
Gal, S.G., Niculescu, C.P., Nonlinear versions of Korovkin’s abstract theorems, Revista de la Real Academia de Ciencias Exactas, Fisicas y Naturales, Serie A, Matematicas 116(2022), Article no. 68.
Grabisch, M., Set Functions, Games and Capacities in Decision Making. Springer, New York, 2016.
Korovkin, P.P., On convergence of linear positive operators in the space of continuous functions, (Russian), Doklady Akad. Nauk. SSSR. (NS), 90(1953), 961-964.
Korovkin, P.P., Linear Operators and Approximation Theory, Fitzmatgiz, Moscow, 1959 (in Russian) English translation, Hindustan Publ. Corp., Delhi, 1960.
Lorentz, G.G., Bernstein Polynomials, Chelsea Publ. Company, New York, N.Y., 1986.
Miller, H.I., A measure theoretical subsequence characterization of statistical convergence, Trans. Amer. Math. Soc., 347(1995), 1811-1819.
Pehlivan, S., Mamedov, M.A., Statistical cluster points and turnpike, Optimization, 48(2000), 93-106.
Šalát, T., On statistically convergent sequences of real numbers, Math. Slovaca, 30(1980), 139-150.
Schoenberg, I.J., The integrability of certain functions and related summability methods, Amer. Math. Monthly, 66(1959), 361-375.
Wang, Z., Klir, G.J., Generalized Measure Theory, Springer-Verlag, New York, 2009.
Zygmund, A., Trigonometric Series, 2nd ed., Cambridge University Press, Cambridge, 1979.
Downloads
Published
How to Cite
Issue
Section
License
Copyright (c) 2023 Studia Universitatis Babeș-Bolyai Mathematica
This work is licensed under a Creative Commons Attribution-NonCommercial-NoDerivatives 4.0 International License.
