Abstract
In this paper, we introduce a generalization of Baskakov operators based on a function ρ. We prove a weighted Korovkin type theorem and compute the rate of convergence via weighted modulus of continuity for these operators. Also we give a Voronovskaya type asymptotic formula.
References
[1] Altomare, F.—Mangino, E.: On a generalization of Baskakov operators, Rev. Roumaine Math. Pures. Appl. 44 (1999), 683–705.Search in Google Scholar
[2] Aral, A.—Gupta, V.: Generalized q-Baskakov operators, Math. Slovaca 61 (2011), 619–634.10.2478/s12175-011-0032-3Search in Google Scholar
[3] Aral, A.—Inoan, D.—Raşa, I.: On the generalized Szász-Mirakyan operators, Results Math. 65 (2014), 441–452.10.1007/s00025-013-0356-0Search in Google Scholar
[4] Bardaro, C.—Mantellini, I.: A Voronovskaya type theorem for general class of discrete operators, Rocky Mountain J. Math. 39 (2009), 1411–1442.10.1216/RMJ-2009-39-5-1411Search in Google Scholar
[5] Baskakov, V.A.: An instance of a sequence of linear positive operators in the space of continuous functions, Dokl. Akad. Nauk SSSR (N.S), 113 (1957), 249–251.Search in Google Scholar
[6] Cárdenas-Morales, D.—Garrancho, P.—Raşa, I.: Asymptotic formulae via a Korovkin type result, Abstr. Appl. Anal. (2012), Art. ID 217464, 12 pp.10.1155/2012/217464Search in Google Scholar
[7] Ditzian, Z.: On global inverse theorems for Szász and Baskakov operators, Canad. J. Math. 31 (1979), 255–263.10.4153/CJM-1979-027-2Search in Google Scholar
[8] Erençin, A.—Başcanbaz-Tunca, G.: Approximation properties of a class of linear positive operators in weighted spaces, C.R. Acad. Bulgare Sci. 63 (2010), 1397–1404.Search in Google Scholar
[9] Feng, G.: Direct and inverse approximation theorems for Baskakov operators with the Jacobi-type weight, Abstr. Appl. Anal. 2011 (2011), Art. ID 101852, 13 pp.10.1155/2011/101852Search in Google Scholar
[10] Gadjiev, A. D.: The convergence problem for a sequence of positive linear operators on unbounded sets and theorems analogous to that of P.P. Korovkin, Dokl. Akad. Nauk SSSR 218 (1974), 1001–1004. Also in Soviet Math. Dokl. 15 (1974), 1433–1436, (in english).Search in Google Scholar
[11] Gadjiev, A. D.: Theorems of the type of P. P. Korovkin’s theorems, Math. Zametki 20 (1976), 781–786 (in russian), Math. Notes 20 (1976), 995–998 (in english).Search in Google Scholar
[12] Holhoş, A.: Quantitative estimates for positive linear operators in weighted spaces, General Math. 16 (2008), 99–110.Search in Google Scholar
[13] İspir, N.: On modified Baskakov operators on weighted spaces, Turkish J. Math. 25 (2001), 355–365.Search in Google Scholar
[14] López-Moreno, A.J.: Weighted simultaneous approximation with Baskakov type operators, Acta Math. Hungar. 104 (2004), 143–151.10.1023/B:AMHU.0000034368.81211.23Search in Google Scholar
[15] Miheşan, V.: Uniform approximation with positive linear operators generated by generalized Baskakov method, Automat. Comput. Appl. Math. 7 (1998), 34–37.Search in Google Scholar
[16] Söylemez-Özden, D.—Başcanbaz-Tunca, G.—Aral, A.: Approximation by complex q-Baskakov operators in compact disk, An. Univ. Oradea Fasc. Mat. 21 (2014), 167–182.Search in Google Scholar
[17] Wafi, A.—Khatoon, S.: On the order of approximation of functions by generalized Baskakov operators, Indian J. Pure Appl. Math. 35 (2004), 347–358.Search in Google Scholar
[18] Wafi, A.—Khatoon, S.: Approximation by generalized Baskakov operators for functions of one and two variables in exponential and polynomial weight spaces, Thai. J. Math. 2 (2004), 53–66.Search in Google Scholar
[19] Wafi, A.—Khatoon, S.: Direct and inverse theorems for generalized Baskakov operators in polynomial weight spaces, An. ştiint. Univ. Al. I. Cuza Iaşi. Mat. (N.S) 50 (2004), 159–173.Search in Google Scholar
[20] Wafi, A.—Khatoon, S.: Convergence and Voronovskaja-type theorems for derivatives of generalized Baskakov operators, Cent. Eur. J. Math. 6 (2008), 325–334.10.2478/s11533-008-0025-9Search in Google Scholar
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