×

zbMATH — the first resource for mathematics

Korovkin-type theorems and applications. (English) Zbl 1179.41024
Summary: 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.

MSC:
41A36 Approximation by positive operators
47B65 Positive linear operators and order-bounded operators
PDF BibTeX XML Cite
Full Text: DOI
References:
[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
[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 · Zbl 1118.41015
[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 · 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
[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
[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 · 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
[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 · Zbl 1088.41008
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.