On Baskakov operators preserving the exponential function

Ovgu Gurel Yilmaz; Vijay Gupta; Ali Aral

doi:10.33993/jnaat462-1110

Authors

Ovgu Gurel Yilmaz Department of Mathematics, Faculty of Sciences, Ankara University, Turkey
Vijay Gupta Department of Mathematics, Netaji Subhas Institute of Technology, Sector 3 Dwarka, New Delhi 110078, India
Ali Aral Department of Mathematics, Faculty of Sciences and Arts, Kırıkkale University, Turkey

DOI:

https://doi.org/10.33993/jnaat462-1110

Keywords:

Baskakov operators, King type operators, Voronovskaya type theorems, modulus of continuity

Abstract views: 535

Abstract

In this paper, we are concerned about the King-type Baskakov operators defined by means of the preserving functions \(e_{0}\) and \(e^{2ax},\ a>0\) fixed.

Using the modulus of continuity, we show the uniform convergence of new operators to \(f\). Also, by analyzing the asymptotic behavior of King-type operators with a Voronovskaya-type theorem, we establish shape preserving properties using the generalized convexity.

Downloads

Download data is not yet available.

References

T. Acar, A. Aral and H. Gonska, On Szasz-Mirakyan operators preserving e2ax, a >0, Mediterranean J. Math., 14 (2017) no. 1, pp. 400-408, https://doi.org/10.1007/s00009-016-0804-7 DOI: https://doi.org/10.1007/s00009-016-0804-7

O. Agratini, Uniform approximation of some classes of linear positive operators expressed by series, Applicable Analysis, 94 (2015) no. 8, pp. 1662-1669, https://doi.org/10.1080/00036811.2014.940919 DOI: https://doi.org/10.1080/00036811.2014.940919

O. Agratini, Approximation properties of a class of linear operators, Math. Methods Appl. Sci., 36 (2013) no. 17, pp. 2353-2358, https://doi.org/10.1002/mma.2758 DOI: https://doi.org/10.1002/mma.2758

O. Agratini and S. Tarabie, On approximating operators preserving certain polynomials, Automat. Comput. Appl. Math.,17(2008) no. 2 , pp. 191-199.

J. M. Aldaz and H. Render, Optimality of generalized Bernstein operators, J. Approx. Theory, 162 (2010) no. 7 , pp. 1407-1416, https://doi.org/10.1016/j.jat.2010.03.003 DOI: https://doi.org/10.1016/j.jat.2010.03.003

A. Aral, D. Inoan and I. Rasa, On the generalized Szasz-Mirakyan operators, Results in Math, 65 (2014) no. 3-4 , pp. 441-452, https://doi.org/10.1007/s00025-013-0356-0 DOI: https://doi.org/10.1007/s00025-013-0356-0

V.A. Baskakov, An example of sequence of linear positive operators in the space of continuous functions, Dokl. Akad. Nauk. SSSR, 113 (1957), pp. 249-251.

P. I. Braica, L I. Piscoran and A. Indrea, Grafical structure of some King type operators , Acta Universitatis Apulensis, (2014) no. 34 , pp. 163-171.

M. Birou, A note about some general King-type operators, Ann. Tiberiu Popoviciu Semin. Funct. Equ. Approx. Convexity, 12 (2014), pp. 3-16.

B. D. Boyanov and V. M. Veselinov, A note on the approximation of functions in an infinite interval by linear positive operators, Bull. Math. Soc. Sci. Math. Roum., 14 (62) (1970) no. 1 , pp. 9-13.

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) no. 2 , 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, https://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) no. 2, pp. 625-638, https://doi.org/10.1016/s0022-247x(03)00048-9 DOI: https://doi.org/10.1016/S0022-247X(03)00048-9

W. Z. Chen, Approximation Theory of Operators, Xiamen University Publishing House, Xiamen, China, 1989.

O. Duman and M. Ali Ozarslan, Szasz- Mirakyan type operators providing a better error estimation, Appl. Math. Lett, 20 (2007) no. 12 , pp. 1184-1188, https://doi.org/10.1016/j.aml.2006.10.007 DOI: https://doi.org/10.1016/j.aml.2006.10.007

V. Gupta and R. P. Agarwal, Convergence Estimates in Approximation Theory, Springer, Cham. 2014, https://doi.org/10.1007/978-3-319-02765-4 DOI: https://doi.org/10.1007/978-3-319-02765-4

V. Gupta, N. Malik and Th. M. Rassias, Moment generating functions and moments of linear positive operators, Modern Discrete Mathematics and Analysis (Edited by N. J. Daras and Th. M. Rassias), Springer 2017. DOI: https://doi.org/10.1007/978-3-319-74325-7_8

V. Gupta and G. Tachev, Approximation with Positive Linear Operators and Linear Combinations, Springer 2017, https://doi.org/10.1007/978-3-319-58795-0 DOI: https://doi.org/10.1007/978-3-319-58795-0

A. Holhos, The rate of approximation of functions in an infinite interval by positive linear operators, Studia Univ. ”Babes-Bolyai”, Mathematica, 55 (2010) no. 2 , pp. 133-142.

N. Ispir, On modified Baskakov operators on weighted spaces, Turk J. Math, 25 (2001) no. 3 , pp. 355-365.

J.P. King, Positive linear operators which preserve x2, Acta Math. Hungar., 99 (2003) no. 3, pp. 203–208. DOI: https://doi.org/10.1023/A:1024571126455

M. A. Ozarslan and H. Altuglu, Local approximation properties for certain King type operators, Filomat, 27 (2013) no. 1 , pp. 173-181, https://doi.org/10.2298/fil1301173o DOI: https://doi.org/10.2298/FIL1301173O

Z. Ziegler, Linear approximation and generalized convexity, J. Approx. Theory,1 (1968) no. 4 , pp. 420-443, https://doi.org/10.1016/0021-9045(68)90031-2 DOI: https://doi.org/10.1016/0021-9045(68)90031-2