In the present paper we introduce and investigate weighted statistical approximation properties of a q-analogue of the Baskakov and Baskakov-Kantorovich operators. By using a weighted modulus of smoothness, we give some direct estimations for error in the case 0 < q < 1.
@article{bwmeta1.element.doi-10_2478_s11533-010-0040-5,
author = {Nazim Mahmudov},
title = {Statistical approximation of Baskakov and Baskakov-Kantorovich operators based on the q-integers},
journal = {Open Mathematics},
volume = {8},
year = {2010},
pages = {816-826},
zbl = {1204.41017},
language = {en},
url = {http://dml.mathdoc.fr/item/bwmeta1.element.doi-10_2478_s11533-010-0040-5}
}
Nazim Mahmudov. Statistical approximation of Baskakov and Baskakov-Kantorovich operators based on the q-integers. Open Mathematics, Tome 8 (2010) pp. 816-826. http://gdmltest.u-ga.fr/item/bwmeta1.element.doi-10_2478_s11533-010-0040-5/
[1] Agratini O., On statistical approximation in spaces of continuous functions, Positivity, 2009, 13(4), 735–743 http://dx.doi.org/10.1007/s11117-008-3002-4[WoS][Crossref] | Zbl 1179.41023
[2] Altomare F., Campiti M., Korovkin-type Approximation Theory and its Applications, de Gruyter Studies in Mathematics, 17, Walter de Gruyter, Berlin-New York, 1994 | Zbl 0924.41001
[3] Andrews G.E., Askey R., Roy R., Special Functions, Cambridge University Press, Cambridge, 1999
[4] Aral A., Gupta V., On the Durrmeyer type modification of the q-Baskakov type operators, Nonlinear Anal., 2010, 72(3–4), 1171–1180 | Zbl 1180.41012
[5] Baskakov V.A., An instance of a sequence of linear positive operators in the space of continuous functions, Dokl. Akad. Nauk. SSSR, 1957, 113, 249–251 (in Russian) | Zbl 0080.05201
[6] Derriennic M.-M., Modified Bernstein polynomials and Jacobi polynomials in q-calculus, Rend. Circ. Mat. Palermo, 2005, 76, 269–290
[7] Doğgru O., Duman O., Statistical approximation of Meyer-König and Zeller operators based on q-integers, Publ. Math. Debrecen, 2006, 68(1–2), 199–214
[8] Doğgru O., Duman O., Orhan C., Statistical approximation by generalized Meyer-König and Zeller type operators, Studia Sci. Math. Hungar., 2003, 40(3), 359–371
[9] Doğgru O., Gupta V., Monotonicity and the asymptotic estimate of Bleimann Butzer and Hahn operators based on q-integers, Georgian Math. J., 2005, 12(3), 415–422
[10] Doğgru O., Gupta V., Korovkin-type approximation properties of bivariate q-Meyer-Konig and Zeller operators, Calcolo, 2006, 43(1), 51–63 http://dx.doi.org/10.1007/s10092-006-0114-8[Crossref] | Zbl 1121.41020
[11] Duman O., Orhan C., Statistical approximation by positive linear operators, Studia Math., 2004, 161(2), 187–197 http://dx.doi.org/10.4064/sm161-2-6[Crossref] | Zbl 1049.41016
[12] Gadjiev A.D., Orhan C., Some approximation theorems via statistical convergence, Rocky Mountain J. Math., 2002, 32(1), 129–138 http://dx.doi.org/10.1216/rmjm/1030539612[Crossref] | Zbl 1039.41018
[13] Gupta V., Some approximation properties of q-Durrmeyer operators, Appl. Math. Comput., 2008, 197(1), 172–178 http://dx.doi.org/10.1016/j.amc.2007.07.056[Crossref][WoS]
[14] Gupta V., Radu C., Statistical approximation properties of q-Baskakov.Kantorovich operators, Cent. Eur. J. Math., 2009, 7(4), 809–818 http://dx.doi.org/10.2478/s11533-009-0055-y[WoS][Crossref] | Zbl 1183.41015
[15] Kac V., Cheung P., Quantum Calculus, Universitext, Springer, New York, 2002
[16] López-Moreno A.-J., Weighted silmultaneous approximation with Baskakov type operators, Acta Math. Hungar., 2004, 104(1–2), 143–151 http://dx.doi.org/10.1023/B:AMHU.0000034368.81211.23[Crossref]
[17] Lupaş A., A q-Analogue of the Bernstein operator, In: Seminar on Numerical and Statistical Calculus, University of Cluj-Napoca, 1987, 85–92 | Zbl 0696.41023
[18] Mahmudov N.I., Korovkin-type theorems and applications, Cent. Eur. J. Math., 2009, 7(2), 348–356 http://dx.doi.org/10.2478/s11533-009-0006-7[Crossref][WoS] | Zbl 1179.41024
[19] Mahmudov N.I., On q-parametric Szász-Mirakjan operators, Mediterranean J. Math., (in press), DOI:10.1007/s00009-010-0037-0 [Crossref] | Zbl 1206.41015
[20] Mahmudov N.I., Sabancıgil P., q-Parametric Bleimann Butzer and Hahn operators, J. Inequal. Appl., 2008, art. ID 816367, 15 pp. [WoS]
[21] Mahmudov N.I., Sabancıgil P., On genuine q-Bernstein.Durrmeyer operators, Publ. Math. Debrecen, 2010, 76(1–2), (in press) | Zbl 1274.41033
[22] Özarslan M.A., q-Szász Schurer operators, (in press)
[23] Phillips G.M., Bernstein polynomials based on the q-integers, Ann. Numer. Math., 1997, 4(1–4), 511–518 | Zbl 0881.41008
[24] Radu C., On statistical approximation of a general class of positive linear operators extended in q-calculus, Appl. Math. Comput., 2009, 215(6), 2317–2325 http://dx.doi.org/10.1016/j.amc.2009.08.023[Crossref][WoS] | Zbl 1179.41025
[25] Trif T., Meyer-König and Zeller operators based on the q-integers, Rev. Anal. Numer. Theor. Approx., 2000, 29(2), 221–229 | Zbl 1023.41022