Mathematica Bohemica, Vol. 143, No. 4, pp. 419-430, 2018


Abstract Korovkin type theorems on modular spaces by $\mathscr{A}$-summability

Emre Taş

Received June 8, 2017.   First published March 26, 2018.

Abstract:  Our aim is to change classical test functions of Korovkin theorem on modular spaces by using $\mathscr{A}$-summability.
Keywords:  $\mathscr{A}$-summability; modular space; abstract Korovkin theory
Classification MSC:  40C05, 41A36


References:
[1] F. Altomare: Korovkin-type theorems and approximation by positive linear operators. Surv. Approx. Theory 5 (2010), 92-164. MR 2721174 | Zbl 1285.41012
[2] F. Altomare, S. Diomede: Contractive Korovkin subsets in weighted spaces of continuous functions. Rend. Circ. Mat. Palermo II. Ser. 50 (2001), 547-568. DOI 10.1007/BF02844431 | MR 1871614 | Zbl 1011.46020
[3] Ö. G. Atlihan, E. Taş: An abstract version of the Korovkin theorem via $\Cal A$-summation process. Acta Math. Hung. 145 (2015), 360-368. DOI 10.1007/s10474-015-0476-y | MR 3325796 | Zbl 1363.41022
[4] C. Bardaro, A. Boccuto, K. Demirci, I. Mantellini, S. Orhan: Korovkin-type theorems for modular $\Psi$-$A$-statistical convergence. J. Funct. Spaces 2015 (2015), Article ID 160401, 11 pages. DOI 10.1155/2015/160401 | MR 3310460 | Zbl 1327.46006
[5] C. Bardaro, A. Boccuto, K. Demirci, I. Mantellini, S. Orhan: Triangular $A$-statistical approximation by double sequences of positive linear operators. Result. Math. 68 (2015), 271-291. DOI 10.1007/s00025-015-0433-7 | MR 3407558 | Zbl 1338.40009
[6] C. Bardaro, A. Boccuto, X. Dimitriou, I. Mantellini: Abstract Korovkin-type theorems in modular spaces and applications. Cent. Eur. J. Math. 11 (2013), 1774-1784. DOI 10.2478/s11533-013-0288-7 | MR 3080236 | Zbl 1283.41018
[7] C. Bardaro, A. Boccuto, X. Dimitriou, I. Mantellini: Modular filter convergence theorems for abstract sampling type operators. Appl. Anal. 92 (2013), 2404-2423. DOI 10.1080/00036811.2012.738480 | MR 3169171 | Zbl 1286.41003
[8] C. Bardaro, I. Mantellini: Korovkin theorem in modular spaces. Ann. Soc. Math. Pol., Ser. I, Commentat. Math. 47 (2007), 239-253. MR 2377960 | Zbl 1181.41035
[9] C. Bardaro, I. Mantellini: Multivariate moment type operators: approximation properties in Orlicz spaces. J. Math. Inequal. 2 (2008), 247-259. DOI 10.7153/jmi-02-22 | MR 2426828 | Zbl 1152.41308
[10] C. Bardaro, I. Mantellini: A Korovkin theorem in multivariate modular function spaces. J. Funct. Spaces Appl. 7 (2009), 105-120. DOI 10.1155/2009/863153 | MR 2541228 | Zbl 1195.41021
[11] C. Bardaro, J. Musielak, G. Vinti: Nonlinear Integral Operators and Applications. De Gruyter Series in Nonlinear Analysis and Applications 9. Walter de Gruyter, Berlin (2003). MR 1994699 | Zbl 1030.47003
[12] H. T. Bell: Order summability and almost convergence. Proc. Am. Math. Soc. 38 (1973), 548-552. DOI 10.2307/2038948 | MR 0310489 | Zbl 0259.40003
[13] A. Boccuto, X. Dimitriou: Modular filter convergence theorems for Urysohn integral operators and applications. Acta Math. Sin., Engl. Ser. 29 (2013), 1055-1066. DOI 10.1007/s10114-013-1443-6 | MR 3048228 | Zbl 1268.41018
[14] K. Demirci, S. Orhan: Statistical relative approximation on modular spaces. Result. Math. 71 (2017), 1167-1184. DOI 10.1007/s00025-016-0548-5 | MR 3648467
[15] S. Karakuş, K. Demirci: Matrix summability and Korovkin type approximation theorem on modular spaces. Acta Math. Univ. Comen., New Ser. 79 (2010), 281-292. MR 2745177 | Zbl 1240.41065
[16] S. Karakuş, K. Demirci, O. Duman: Statistical approximation by positive linear operators on modular spaces. Positivity 14 (2010), 321-334. DOI 10.1007/s11117-009-0020-9 | MR 2657637 | Zbl 1193.41014
[17] J. Musielak: Orlicz Spaces and Modular Spaces. Lecture Notes in Mathematics 1034. Springer, Berlin (1983). DOI 10.1007/BFb0072210 | MR 0724434 | Zbl 0557.46020
[18] J. Musielak, W. Orlicz: On modular spaces. Studia Math. 18 (1959), 49-65. DOI 10.4064/sm-18-1-49-65 | MR 0101487 | Zbl 0086.08901
[19] H. Nakano: Modulared Semi-Ordered Linear Spaces. Tokyo Math. Book Series, Vol. 1. Maruzen, Tokyo (1950). MR 0038565 | Zbl 0041.23401
[20] S. Orhan, K. Demirci: Statistical ${\scr A}$-summation process and Korovkin type approximation theorem on modular spaces. Positivity 18 (2014), 669-686. DOI 10.1007/s11117-013-0269-x | MR 3275359 | Zbl 1308.41024
[21] I. Sakaoğlu, C. Orhan: Strong summation process in $L_p$ spaces. Nonlinear Anal., Theory Methods Appl., Ser. A, Theory Methods 86 (2013), 89-94. DOI 10.1016/j.na.2013.03.010 | MR 3053558 | Zbl 1283.41017

Affiliations:   Emre Taş, Department of Mathematics, Ahi Evran University, Bağbaşi Mahallesi, Şht. Sahir Kurutluoğlu Cd., 40100 Merkez/Kirşehir, Turkey, e-mail: emretas86@hotmail.com


 
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