Remarks on a Bernstein-type operator of Aldaz, Kounchev and Render

Authors

  • Ana Maria Acu Lucian Blaga University of Sibiu
  • Heiner Gonska
  • Margareta Heilmann

Keywords:

Bernstein-type operator, king operator, second order modulus of continuity, Marsden-Schoenberg, modulus of order j

Abstract

The Bernstein-type operator of Aldaz, Kounchev and Render (2009) is discussed. New direct results in terms of the classical second order modulus as well as in a modification following Marsden and Schoenberg are given.

Downloads

Download data is not yet available.

References

A.M. Acu, I. Rasa, New estimates for the differences of positive linear operators, Numer. Algor.,73(2016) no. 3, pp. 775–789, https://doi.org/10.1007/s11075-016-0117-8.

O. Agratini, I.A. Rus, Iterates of a class of discrete linear operator via contraction principle, Comment. Math. Univ. Carol., 44(2003) 3, 555–563.

J.M. Aldaz, O. Kounchev, H. Render, Shape preserving properties of generalizedBernstein operators on extended Chebyshev spaces, Numer. Math.,114(2009) no. 1, pp. 1–25, https://doi.org/10.1007/s00211-009-0248-0.

J.M. Aldaz, H. Render, Generalized Bernstein operators on the classical polynomial spaces, Mediterr. J. Math.,15(2018) art. id. 222, https://doi.org/10.1007/s00009-018-1266-x.

M. Birou, A proof of a conjecture about the asymptotic formula of a Bernstein typeoperator, Results Math.,72(2017), pp. 1129–1138, https://doi.org/10.1007/s00025-016-0608-x.

D. Cardenas-Morales, P. Garrancho, F.J. Munoz-Delgado, Shape preservingapproximation by Bernstein-type operators which fix polynomials, Appl. Math. Comput.,182(2006), 1615–1622, https://doi.org/10.1016/j.amc.2006.05.046.

D. Cardenas-Morales, P. Garrancho, I. Rasa, Bernstein-type operators whichpreserve polynomials, Comput. Math. Appl.,62(2011), 158-163, https://doi.org/10.1016/j.camwa.2011.04.063.

D. Cardenas-Morales, P. Garrancho, I. Rasa, Asymptotic formulae via a Korovkin-type result, Abstract Appl. Anal.,2012(2021), art. ID 217464, 12 pp., https://doi.org/10.1155/2012/217464.

C. Cottin, I. Gavrea, H. Gonska, D. Kacso, Ding-Xuan Zhou, Global smoothness preservation and the variation-diminishing property, J. Inequal. Appl.,4(1999) no. 2, 91–114, https://doi.org/10.1155/s1025583499000314.

Z. Finta, Bernstein type operators having1andxjas fixed points, Cent. Eur. J. Math., 11(2013), 2257–2261.

Z. Finta, Note on a Korovkin-type theorem, J. Math. Anal. Appl., 415 (2014), 750–759, https://doi.org/10.1016/j.jmaa.2014.02.010.

Z. Finta, A quantitative variant of Voronovskaja’s theorem for King-type operators,Constructive Mathematical Analysis,2(2019) no. 3, 124–129, https://doi.org/10.33205/cma.553427.

I. Gavrea, M. Ivan, Asymptotic behaviour of the iterates of positive linear operators, Abstract Appl. Anal., 2011(2011), art. ID 670509, 11 pp., https://doi.org/10.1155/2011/670509.

H. Gonska, On approximation by linear operators: improved estimates, Rev. Anal.Numer. Theor. Approx., 14(1985), 7-32, https://ictp.acad.ro/jnaat/journal/article/view/1985-vol14-no1-art2.

H. Gonska, P. Pitul, Remarks on an article of J.P. King, Comment. Math. Univ.Carol., 46(2005), 645–652.

J.P. King, Positive linear operators which preservex2, Acta Math. Hungar., 99(2003)no. 3, 203–208, https://doi.org/10.1023/a:1024571126455.

M. Marsden, I.J. Schoenberg, On variation diminishing spline approximation methods, Mathematica, 8 (31)(1966) no. 1, 61–82.

R. Paltanea, Optimal constant in approximation by Bernstein operators, J. Comput.Anal. Appl., 5(2003), 195–235.

R. Paltanea, Approximation Theory using Positive Linear Operators, Birkhauser, Boston, 2004.

Downloads

Published

2021-11-19

How to Cite

Acu, A. M., Gonska, H., & Heilmann, M. (2021). Remarks on a Bernstein-type operator of Aldaz, Kounchev and Render. J. Numer. Anal. Approx. Theory, 50(1), 3–11. Retrieved from https://ictp.acad.ro/jnaat/journal/article/view/1237

Issue

Section

Articles