On Szász-Mirakyan type operators preserving polynomials

Ovgu Gurel Yilmaz; Ali Aral; Fatma Taşdelen Yeşildal

doi:10.33993/jnaat461-1087

Authors

Ovgu Gurel Yilmaz University of Ankara, Department of Mathematics, Turkey
Ali Aral University of Kırıkkale, Department of Mathematics, Turkey
Fatma Taşdelen Yeşildal University of Ankara, Department of Mathematics, Turkey

DOI:

https://doi.org/10.33993/jnaat461-1087

Keywords:

Szász-Mirakyan operators, shape preserving properties, Voronovskaya-type theorem., modified operator

Abstract views: 498

Abstract

In this paper, a modification of Szász-Mirakyan operators is studied [1] which generalizes the Szász-Mirakyan operators with the property that the linear combination \(e_2 + \alpha e_1\) of the Korovkin's test functions \(e_1\) and \(e_2\) are reproduced for \(\alpha\geq 0\). After providing some computational results, shape preserving properties of mentioned operators are obtained. Moreover, some estimations for the rate of convergence of these operators by using different type modulus of continuity are shown. Furthermore, a Voronovskaya-type formula and an approximation result for derivative of operators are calculated. Finally, some graphics which are based on our main results are shown.

Downloads

Download data is not yet available.

Author Biographies

Ovgu Gurel Yilmaz, University of Ankara, Department of Mathematics, Turkey

MATHS DEPARTMENT

Ali Aral, University of Kırıkkale, Department of Mathematics, Turkey

MATS DEPARTMENT

Fatma Taşdelen Yeşildal, University of Ankara, Department of Mathematics, Turkey

MATHS DEPARTMENT

References

O. Agratini and S. Tarabie, On approximating operators preserving certain polynomials, Automat. Comput. Appl. Math., 17 (2008), pp. 191-199.

A. Aral, D. Inoan and I. Rasa, On the generalized Szasz-Mirakyan operators, Results in Math., 65 (2014), pp. 441-452, https://doi.org/10.1007/s00025-013-0356-0 DOI: https://doi.org/10.1007/s00025-013-0356-0

P. I. Braica, L I. Piscoran and A. Indrea, Graphical lectures of some King type operators, Acta Univ. Apulensis Math. Inform., 34 (2013), pp. 163-171.

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

D. Cardenas-Morales, P. Garrancho and I. Rasa, Approximation properties of Bernstein-Durrmeyer type operators, Appl. Math. Comput., 232 (2014), pp. 1-8, http://dx.doi.org/10.1016/j.amc.2014.01.046 DOI: https://doi.org/10.1016/j.amc.2014.01.046

J. de la cal and J. Carcamo, On uniform approximation by some classical Bernstein-type operators, J. Math. Anal. Appl, 279 (2003), pp. 625-638, https://doi.org/10.1016/S0022-247X(03)00048-9 DOI: https://doi.org/10.1016/S0022-247X(03)00048-9

O. Duman and M. Ali Ozarslan, Sz ́asz-Mirakyan type operators providing a better error estimation, Appl. Math. Lett., 20 (2007), pp. 1184-1188, http://dx.doi.org/10.1016/j.aml.2006.10.007 DOI: https://doi.org/10.1016/j.aml.2006.10.007

N. Ispir, On modified Baskakov operators on weighted spaces, Turkish J. Math, 25 (2001), pp. 355-365.

J.P. King, Positive linear operators which preserve x2, Acta Math. Hungar, 99 (2003), pp. 203-208, https://doi.org/10.1023/a:1024571126455 DOI: https://doi.org/10.1023/A:1024571126455

O. Duman and M. Ali Ozarslan, Local approximation properties for certain King type operators, Filomat, 27 (2013), pp. 173-181, https://doi.org/10.2298/FIL1301173O DOI: https://doi.org/10.2298/FIL1301173O

G. M. Phillips, Interpolation and Approximation by Polynomials, Springer-Verlag, New York, 2003, https://doi.org/10.1007/b97417 DOI: https://doi.org/10.1007/b97417

D. D. Stancu, Approximation of functions by a new class of linear polynomial operators, Rev. Roumanine Math. Pures Appl., 13 (1968), pp. 1173-1194.

K. G. Weierstrass, Uber Die Analytische Darstel lbarkeit Sogenannter Wil lkurlicher Funktionen Einer Reel len Veranderlichen, Sitzungsber. Akad. Berlin, pp. 633-639, 789-805, 1885.

Z. Ziegler, Linear approximation and generalized convexity , J. Approx. Theory, 1 (1968), pp. 420-443, https://doi.org/10.1016/0021-9045(68)90031-2 DOI: https://doi.org/10.1016/0021-9045(68)90031-2