Summation process of monotone and sublinear operators in B−statistical sense
DOI:
https://doi.org/10.24193/subbmath.2026.1.09Keywords:
B−statistical convergence, monotone operator, sublinear operator, Korovkin-type theoremsAbstract
By employing the A−summation process in the B−statistical sense, where A and B are sequences of infinite matrices, we provide new results on the classical Korovkin theorem for a sequence of monotone and sublinear operators. Reported results essentially extend some theorems existing in the literature.
References
[1] Altomare, F, Campiti, M., Korovkin Type Approximation Theory and its Applications, Walter de Gruyter Publ., Berlin, 1994.
[2] Anastassiou, G.A., Approximations by sublinear operators, Acta Math. Univ. Comenian. (N.S.) 87 (2018), no. 2, 237–250.
[3] Atlıhan, ¨O.G., Orhan C., Matrix summability and positive linear operators, Positivity 11 (2007), 387–398.
[4] Atlıhan, ¨O.G., Orhan C., Summation process of positive linear operators, Computers & Mathematics with Applications, 56 (2008), no. 5, 1188-1195.
[5] Bal´aˇz, V., ˇSal´at, T., Uniform density u and corresponding I_u−convergence, Math. Commun. 11 (2006), no. 1, 1–7.
[6] Balcerzak, M., Dems, K., Komisarski, A., Statistical convergence and ideal convergence for sequences of functions, Journal of Mathematical Analysis and Applications, 328 (2007), no. 1, 715-729.
[7] Bell, H.T., A General Method of Summability: A−summability, Lehigh University Phd Dissertation & Thesis 1971 https://preserve.lehigh.edu/lehigh-scholarship/graduate-publications-theses-dissertations/theses-dissertations/general-method
[8] Bell, H.T., Order summability and almost convergence, Proc. Amer. Math. Soc., 38(1973), 548–552.
[9] Bojanic, R., Khan, M.K., Summability of hermite-fejer interpolation for functions of bounded variation, J Nat Sci Math 32 (1992), 5–10.
[10] Costin, O., Dunne, G.V., Convergence from divergence, J. Phys. A, 51 (2018), no. 4, 10 pp.
[11] C¸ ınar, S., Yıldız, S., P −statistical summation process of sequences of convolution operators, Indian Journal of Pure and pplied Mathematics, 53 (2022), no. 3, 648-659.
[12] Demirci, K., Dirik, F., Yıldız, S., Approximation Results via Power Series Method for Sequences of Monotone and Sublinear Operators. Results Math., 80(2025), no. 2, 1-19. DOI: https://doi.org/10.1007/s00025-025-02366-w
[13] Demirci, K., Dirik, F., Yıldız, S., Weighted Convergence and Applications to Approximation Theorems for Sequences of Monotone and Sublinear Operators. Iran. J. Sci., (2025) 1-9. DOI: https://doi.org/10.1007/s40995-025-01839-5
[14] Dirik, F., Demirci, K., Korovkin type approximation theorems in B−statistical sense, Mathematical and computer modelling, 49 (2009), no. 9-10, 2037-2044.
[15] Fast, H., Sur la convergence statistique, Colloquium Mathematicae 2 (1951), 241–244.
[16] Fridy, J.A., On statistical convergence, Analysis 5 (1985), 301-313.
[17] Gadjiev, A.D, Orhan, C., Some approximation theorems via statistical convergence, Rocky Mt. J. Math. 32 (2002), 129-138.
[18] Gal, S.G., Iancu, I.T., Korovkin-Type theorems for statistically convergent sequences of monotone and sublinear operators, Bull. Malays. Math. Sci. Soc. 46 (2023), no .2.
[19] Gal, S.G., Niculescu, C.P., A nonlinear extension of Korovkin’s theorem, Mediterr. J. Math., 17 (2020), no. 5.
[20] Gal, S.G., Niculescu, C.P., A note on the Choquet type operators, Aequ. Math. 95 (2021), 433–447.
[21] Gal, S.G., Niculescu, C.P., Nonlinear versions of Korovkin’s abstract theorems, Rev. Real Acad. Cienc. Exactas Fis. Nat. Ser. A-Mat., 116 (2022), no .2.
[22] Iancu, I.T., Statistical Korovkin-type theorem for monotone and sublinear operators, Studia Universitatis Babes-Bolyai, Mathematica, 68 (2023), no. 2.
[23] Kolk, E., Inclusion relations between the statistical convergence and strong summability, Acta Commun. Univ. Tartu. Math. 2 (1998), 39–54.
[24] King, J.P., Swetits, J.J., Positive linear operators and summability, J. Aust. Math. Soc. 11 (1970), 281–290.
[25] Korovkin, P.P., Linear Operators and the Theory of Approximation, Hindustan Publ. Co. Delhi, 1960.
[26] Lorentz, G.G., A contribution to the theory of divergent sequences, Acta Math. 80 (1948), 167–190.
[27] O˘guz, G., G¨ulfırat, M., Mean Ergodic type theorems by means of ideal convergence Advances in Operator Theory, 10 (2025), no. 3, 1-11.
[28] O˘guz, G., Orhan, C., Poisson’s Equation for A−Mean Ergodic Operators, Results in Mathematics, 77 (2022), no.1, 53.
[29] O˘guz, G., Orhan, C., Some mean and uniform ergodic type theorems Filomat, 36 (2022), no.7, 2403-2410.
[30] Orhan, S., Dirik, F., Demirci, K., A Korovkin-type approximation theorem for double
sequences of positive linear operators of two variables in statistical A−summability sense, Miskolc Mathematical Notes, 5 (2014), no. 2, 625-633.
[31] Orhan, S., Acar, T., Dirik, F., Korovkin type theorems in weighted Lp−spaces via statistical A-summability, An. S¸tiint¸. Univ. Al. I. Cuza Ia¸si. Mat. (N. S.) 62 (2016), no. 2, 537–546.
[32] Orhan, S., Demirci, K., Statistical approximation by double sequences of positive linear operators on modular spaces, Positivity, 18 (2014), pp. 669-686.
[33] Pehlivan, S., Strongly almost convergent sequences defined by a modulus and uniformly statistical convergence, Soochow J. Math. 20 (1994), no. 2, 205–211.
[34] Sakao˘glu, ˙I., Orhan, C., Strong summation process in Lp spaces, Nonlinear Analysis, 86 (2013), 89-94.
[35] S¨oylemez, D., ¨Unver, M., Korovkin type theorems for Cheney–Sharma Operators via summability methods, Results Math. 72 (2017), 1601–1612.
[36] Steinhaus, H., Sur la convergence ordinaire et la convergence asymtotique, Colloq. Math. 2 (1951) 73–74.
[37] Stieglitz, M., Eine verallgenmeinerung des begriffs festkonvergenz, Math. Japonica 18 (1973) 53–70.
[38] S¸ahin Bayram, N., Orhan, C., A−Summation process in the space of locally integrable functions, Stud. Univ. Babes-Bolyai Math., 65 (2020), no. 2, 255-268.
[39] S¸ahin Bayram, N., Orhan, C., Abel Convergence of the Sequence of Positive Linear Operators in Lp,q (loc), Bull. Belg. Math. Soc. Simon Stevin 26 (2019), no. 1, 71–83.
[40] S¸ahin Bayram, N., Strong summation process in locally integrable function spaces, Hacettepe Journal of Mathematics and Statistics, 45 (2016), no. 3, 683-694.
[41] ¨Unver, M., Abel transforms of positive linear operators on weighted spaces, Bull. Belg. Math. Soc. 21 (2014), no. 5, 813–822.
[42] Yıldız, S., Abstract versions of Korovkin theorems on modular spaces via statistical relative summation process for double sequences, Tbilisi Mathematical Journal, 13 (2020), no. 1, 139-156.
[43] Yıldız, S., Demirci, K., Dirik, F., Approximation theorems via power series statistical convergence and applications for sequences of monotone and sublinear operators, Real Acad. Cienc. Exactas Fis. Nat. Ser. A-Mat., 119 (2025), no. 4, 100. DOI: https://doi.org/10.1007/s13398-025-01767-4
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