A Note On Statistical Approximation Properties of Complex q-Szász- Mirakjan Operators

Abstract

The complex q-Szász-Mirakjan operator attached to analytic functions satisfying a suitable exponential type growth condition has been studied in [14]. In this paper, we consider the A-statistical convergence of complex q-Szász- Mirakjan operator.

Keywords

• A-Statistical approximation,complex q-Szász- Mirakjan operators,q-exponential functions,q-derivative

References

1. Gadjiev A.D. and Orhan, C., Some approximation theorems via statistical convergence, Rocky Mt. J. Math., vol. 32, no. 1, pp. 129-138, 2002.
2. Connor, J. S., On strong matrix summability with respect to a modulus and statistical convergence, Canad. Math. Bull. 32 (1989), 194-198.
3. Duman O. and Orhan, C., Statistical Approximation in the space of locally integrable functions, Publ. Math., vol.63, no. 1-2, pp. 133-144, 2003.
4. Duman O. and Orhan, C., Rates of a-statistical convergence of operators in the space of locally integrable functions, Appl. Math. Lett., vol.21, no.5, pp.431-435, 2008.
5. Dirik, F., Duman O. and Demirci, K., Approximation in statistical sense to B-continuous functions by positive linear operators, Studia Scientiarum Mathematicarum Hungarica 47 (2010) 289-298.
6. Erkuş E. and Duman, O., A Korovkin type approximation theorem in statistical sense, Studia Sci. Math. Hungarica 43 (2006), 285-294.
7. Sakaoğlu İ. and Ünver, M., Statistical Approximation for multivariable integrable functions, Miskolc Math. Notes, vol.13, no.2, pp. 485-491, 2012.
8. Duman, O., Özarslan M.A. and Doğru, O., On integral type generalizations of positive linear operators, Studia Math. 174 (2006), 1-12.

9. Freedman A. R. and Sember, J. J., Densities and summability , Pacific J. Math. 95 (1981), 293-3005.
10. Fast, H., Sur la convergence statistique, Colloq. Math. 2 (1951), 241-244.
11. Miller, H. I., A measure theorretical subsequence characterization of statistical convergence, Trans. Amer. Math. Soc. 347 (1995), 1811-1819.
12. Aral, A., A generalization of Szász-Mirakjan operators based on q-integer, Mathematical and Computer Modelling 47, (2008), 1052-1062.
13. Aral A. and Duman, O., A Voronovskaya-type formula for SMK operators via statistical convergence, Mathematica Slovaca 61 (2011) 235-244
14. Aydın, D., On Complex q-Szàsz-Mirakjan Operators Commun. Fac.Sci. Univ. Ank. Series A1. Volume 61, Number 2, (2012), 51-66.
15. Gasper, G. and Rahman, M., Basic Hypergeometric Series, Cambridge University Press, Cambridge, 1990.
16. Ernst, T., The history of q-calculus and a new method, U.U.D.M Report 2000, 16, Department of Mathematics, Upsala University.
17. Gal, S. G., Approximation and geometric properties of complex Favard-Szász-Mirakjan operators in compact diks, Comput. Math. Appl., 56, (2008), 1121-1127.
18. Gal, S. G., Approximation by Complex Bernstein and Convolution Type Operators, World Scientific Publishing Co, USA, 2009.
19. Söylemez, D. and Unver, M., Korovkin Type Theorems for Cheney-Sharma Operators via Summability Methods, Results in Mathematics, Volume 72, (2017),1601-1612.
20. Mahmudov, N. I., Approximation properties of complex q-Szász-Mirakjan operators in compact disks. Computers & Mathematics with Applications, (2010), 1784-1791.
21. Phillips, G. M., Interpolation and Approximation by Polynomials, Springer-Verlag, 2003.