Let {T n} be a sequence of linear operators on C[0,1], satisfying that {T n (e i)} converge in C[0,1] (not necessarily to e i) for i = 0,1,2, where e i = t i. We prove Korovkin-type theorem and give quantitative results on C 2[0,1] and C[0,1] for such sequences. Furthermore, we define King’s type q-Bernstein operator and give quantitative results for the approximation properties of such operators.
@article{bwmeta1.element.doi-10_2478_s11533-009-0006-7,
author = {Nazim Mahmudov},
title = {Korovkin-type theorems and applications},
journal = {Open Mathematics},
volume = {7},
year = {2009},
pages = {348-356},
zbl = {1179.41024},
language = {en},
url = {http://dml.mathdoc.fr/item/bwmeta1.element.doi-10_2478_s11533-009-0006-7}
}
Nazim Mahmudov. Korovkin-type theorems and applications. Open Mathematics, Tome 7 (2009) pp. 348-356. http://gdmltest.u-ga.fr/item/bwmeta1.element.doi-10_2478_s11533-009-0006-7/
[1] DeVore R.A., Lorentz G.G., Constructive approximation, Springer, Berlin, 1993 | Zbl 0797.41016
[2] Doğru O., Gupta V., Korovkin-type approximation properties of bivariate q-Meyer-König and Zeller operators, Calcolo, 2006, 43, 51–63 http://dx.doi.org/10.1007/s10092-006-0114-8[Crossref] | Zbl 1121.41020
[3] Gonska H., Pițul P., Remarks on an article of J.P. King, Comment. Math. Univ. Carolin., 2005, 46, 645–652 | Zbl 1121.41013
[4] Heping W., Korovkin-type theorem and application, J. Approx. Theory, 2005, 132, 258–264 http://dx.doi.org/10.1016/j.jat.2004.12.010[Crossref]
[5] Heping W., XueZhi W., Saturation of convergence for q-Bernstein polynomials in the case q ≥ 1, J. Math. Anal. Appl., 2008, 337, 744–750 http://dx.doi.org/10.1016/j.jmaa.2007.04.014[Crossref][WoS] | Zbl 1122.33014
[6] Il’inskii A., Ostrovska S., Convergence of generalized Bernstein polynomials, J. Approx. Theory, 2002, 116, 100–112 http://dx.doi.org/10.1006/jath.2001.3657[Crossref]
[7] King J.P., Positive linear operators which preserve x 2, Acta. Math. Hungar., 2003, 99, 203–208 http://dx.doi.org/10.1023/A:1024571126455[Crossref] | Zbl 1027.41028
[8] Lupaș A., A q-analogue of the Bernstein operator, University of Cluj-Napoca, Seminar on numerical and statistical calculus, 1987, 85–92 | Zbl 0696.41023
[9] Muñoz-Delgado F.J., Cárdenas-Morales D., Almost convexity and quantitative Korovkin type results, Appl. Math. Lett., 1998, 11, 105–108 http://dx.doi.org/10.1016/S0893-9659(98)00065-2[Crossref] | Zbl 0942.41013
[10] Ostrovska S., q-Bernstein polynomials and their iterates, J. Approx. Theory, 2003, 123, 232–255 http://dx.doi.org/10.1016/S0021-9045(03)00104-7[Crossref]
[11] Ostrovska S., The first decade of the q-Bernstein polynomials: results and perspectives, Journal of Mathematical Analysis and Approximation Theory, 2007, 2, 35–51 | Zbl 1159.41301
[12] Phillips G.M., Bernstein polynomials based on the q-integers, Ann. Numer. Math., 1997, 4, 511–518 | Zbl 0881.41008
[13] Phillips G.M., Interpolation and approximation by polynomials, Springer-Verlag, New York, 2003 | Zbl 1023.41002
[14] Trif T., Meyer-König and Zeller operators based on the q-integers, Rev. Anal. Numer. Theory Approx., 2000, 29, 221–229
[15] Videnskii V.S., On some classes of q-parametric positive linear operators, Oper. Theory Adv. Appl., 2005, 158, 213–222 http://dx.doi.org/10.1007/3-7643-7340-7_15[Crossref] | Zbl 1088.41008