Approximation by Baskakov-Szász-Stancu Operators Preserving Exponential Functions

Abstract

The purpose of this paper is to construct a general class of operators which has known Baskakov-Szász-Stancu that preserving constant and $e^{2ax}, a>0$ functions. We scrutinize a uniform convergence result and analyze the asymptotic behavior of our operators, as well. Finally, we discuss the convergence of corresponding sequences in exponential weighted spaces and make a comparison about which one approximates better between classical Baskakov-Szász-Stancu operators and the recent operators.

Keywords

• Exponential functions,Quantitative results,Weighted approximation,Baskakov-Szász-Stancu operators,Weighted approximation

References

1. [1] T. Acar, A. Aral, H. Gonska: On Szász-Mirakyan operators preserving $e^{2ax}$, $a>0$, a > 0. Mediterr. J. Math. 14 (1) (2017), Art. 6, 14 pp.
2. [2] T. Acar, A. Aral, D. Cárdenas-Morales, P. Garrancho: Szász-Mirakyan type operators which fix exponentials. Results Math. 72 (2) (2017), no. 3, 1393-1404.
3. [3] A. Aral, D. Cárdenas-Morales, P. Garrancho: Bernstein-type operators that reproduce exponential functions. J. Math. Inequal.(Accepted).
4. [4] M. Birou: A note about some general King-type operators. Ann. Tiberiu Popoviciu Semin. Funct. Equ. Approx. Convexity, 12 (2014), 3-16.
5. [5] P. I. Braica, L. I. Pi¸scoran, A. Indrea: Grafical structure of some King type operators. Acta Universitatis Apulensis. 34 (2013), 163-171.
6. [6] D. Cárdenas-Morales, P. Garrancho, F. J. Munoz-Delgado: Shape preserving approximation by Bernstein-type operators which fix polynomials. Appl. Math. and Comput. 182 (2) (2006), 1615-1622.
7. [7] D. Cárdenas-Morales, P. Garrancho, I. Ra¸sa: Approximation properties of Bernstein-Durrmeyer type operators. Appl. Math. Comput., 232 (2014), pp. 1-8.
8. [8] E. Deniz, A. Aral, V. Gupta: Note on Szász-Mirakyan-Durrmeyer operators preserving $e^{2ax}$, $a>0$. Numer. Funct. Anal. Optim. 39 (2) (2018), 201-207.

9. [9] O. Duman, M. A. Özarslan: Szász- Mirakyan type operators providing a better error estimation. Appl. Math. Lett. 20 (2007) 1184-1188.
10. [10] A. D. Gadziev: Theorems of the type of P. P. Korovkin’s theorems. Mat Zametki 20 (5), (1976), 781–786.
11. [11] V. Gupta, G.S. Srivastava: Simultaneous approximation by Baskakov-Szász type operators. Bull. Math. Soc. Sci. 37 (85) (1993), 73–85.
12. [12] V. Gupta, A. M. Acu: On Baskakov-Szász-Mirakyan type operators preserving exponential type functions. Positivity. pp 1-11, doi:10.1007/s11117-018-0553-x.
13. [13] V. Gupta, A. Aral: A note on Szász-Mirakyan-Kantorovich type operators preserving $e^{-x}$. Positivity. 22, pp 415-423, doi:10.1007/s11117-017-0518-5.
14. [14] V. Gupta, G. Tachev: On approximation properties of Phillips operators preserving exponential functions. Mediterr. J. Math. 14 (4) (2017), Art. 177, pp. 12.
15. [15] A. Holho¸s: The rate of approximation of functions in an infinite interval by positive linear operators. Studia Univ. Babes- Bolyai Mathematica. 55 (2) (2010), pp. 133-142.
16. [16] J. P. King: Positive linear operators which preserve $x^{2}$. Acta Math. Hungar. 99 (3) (2003) 203-208.
17. [17] V. N. Mishra, M. Mursaleen, P. Sharma: Some approximation properties of Baskakov-Szász- Stancu operators. Appl. Math. Inf. Sci. 9 (6) (2015), 3159-3167.
18. [18] Ö. G. Yılmaz, A. Aral, F. Ta¸sdelen Ye¸sildal: On Szász-Mirakyan type operators preserving polynomials. J. Numer. Anal. Approx. Theory 46 (1) (2017), 93-106.
19. [19] Ö. G. Yılmaz, V. Gupta, A. Aral: On Baskakov operators preserving the exponential function. J. Numer. Anal. Approx. Theory 46 (2) (2017), 150-161.