Korovkin-type theorems and applications

Nazim Mahmudov

Open Mathematics (2009)

  • Volume: 7, Issue: 2, page 348-356
  • ISSN: 2391-5455

Abstract

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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.

How to cite

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Nazim Mahmudov. "Korovkin-type theorems and applications." Open Mathematics 7.2 (2009): 348-356. <http://eudml.org/doc/269642>.

@article{NazimMahmudov2009,
abstract = {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.},
author = {Nazim Mahmudov},
journal = {Open Mathematics},
keywords = {Korovkin approximation; Positive operator; q-Bernstein operators; King’s type q-Bernstein operator; q-operators; positive operator; -Bernstein operators; King’s type -Bernstein operator; -operators},
language = {eng},
number = {2},
pages = {348-356},
title = {Korovkin-type theorems and applications},
url = {http://eudml.org/doc/269642},
volume = {7},
year = {2009},
}

TY - JOUR
AU - Nazim Mahmudov
TI - Korovkin-type theorems and applications
JO - Open Mathematics
PY - 2009
VL - 7
IS - 2
SP - 348
EP - 356
AB - 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.
LA - eng
KW - Korovkin approximation; Positive operator; q-Bernstein operators; King’s type q-Bernstein operator; q-operators; positive operator; -Bernstein operators; King’s type -Bernstein operator; -operators
UR - http://eudml.org/doc/269642
ER -

References

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  1. [1] DeVore R.A., Lorentz G.G., Constructive approximation, Springer, Berlin, 1993 Zbl0797.41016
  2. [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] Zbl1121.41020
  3. [3] Gonska H., Pițul P., Remarks on an article of J.P. King, Comment. Math. Univ. Carolin., 2005, 46, 645–652 Zbl1121.41013
  4. [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. [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] Zbl1122.33014
  6. [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. [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] Zbl1027.41028
  8. [8] Lupaș A., A q-analogue of the Bernstein operator, University of Cluj-Napoca, Seminar on numerical and statistical calculus, 1987, 85–92 Zbl0696.41023
  9. [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] Zbl0942.41013
  10. [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. [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 Zbl1159.41301
  12. [12] Phillips G.M., Bernstein polynomials based on the q-integers, Ann. Numer. Math., 1997, 4, 511–518 Zbl0881.41008
  13. [13] Phillips G.M., Interpolation and approximation by polynomials, Springer-Verlag, New York, 2003 Zbl1023.41002
  14. [14] Trif T., Meyer-König and Zeller operators based on the q-integers, Rev. Anal. Numer. Theory Approx., 2000, 29, 221–229 
  15. [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] Zbl1088.41008

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