SOME APPROXIMATION PROPERTIES OF KANTOROVICH VARIANT OF CHLODOWSKY OPERATORS BASED ON q INTEGER

Abstract

In this paper, we introduce two diğerent Kantorovich type generalization of the q Chlodowsky operators. For the first operators we give some weighted approximation theorems and a Voronovskaja type theorem. Also, we present the local approximation properties and the order of convergence forunbounded functions of these operators . For second operators, we obtain aweighted statistical approximation property

Keywords

• q Chlodowsky operators, modulus of continuity, local approximation,Peetre’s K-functional, statistical convergence

References

1. Aral A., Gupta V., Agarwal R. P., Applications of q Calculus in Operator Theory, Springer, New York, 2013.
2. Kac V., Cheung P., Quantum Calculus, Springer, New York, 2002.
3. Karsli H., Gupta V., Some approximation properties of q Chlodowsky operators, Appl. Math. Comput. (2008), 195, 220–229.
4. Karaisa A., Approximation by Durrmeyer type Jakimoski–Leviatan operators, Math. Meth- ods Appl. Sci. (2015), In Press, DOI 10.1002/mma.3650.
5. Karaisa A., Tollu D. T., Asar Y., Stancu type generalization of q-Favard-Szàsz operators, Appl. Math. Comput. (2015), 264, 249–257.
6. Aral A., Gupta V., Generalized q Baskakov operators, Math.Slovaca (2011), 61, 619–634.
7. Aral A., A generalization of Szàsz-Mirakyan operators based on q integers, Math. Comput. Model. (2008), 47, 1052–1062.
8. Gadjieva E. A., ·Ibikli E., Weighted approximation by Bernstein-Chlodowsky polynomials, Indian J. Pure. Appl. Math. (1999), 30 (1), 83–87.

9. Büyükyazıcı ·I., Approximation by Stancu–Chlodowsky polynomials, Comput. Math. Appl. (2010), 59, 274–282.
10. Yüksel ·I., Dinlemez Ü., Voronovskaja type approximation theorem for q Szàsz–Beta opera- tors, Appl. Math. Comput. (2014), 235, 555–559.
11. Jackson F.H., On the q de…nite integrals, Quart. J. Pure Appl. Math. (1910), 41, 193–203.
12. Altomare F., Campiti M., Korovkin-type Approximation Theory and its Applications. Vol. Walter de Gruyter, 1994.
13. Dalmano¼glu Ö., Do¼gru O., On statistical approximation properties of Kantorovich type qBernstein operators, Math. Comput. Model. (2010), 52, 760–771.
14. Fast H., Sur la convergence statistique, Colloq Math. (1951), 2, 241–244.
15. Gadjiev A.D., Theorems of the type of P. P. Korovkin type theorems, Math. Zametki. (1976) , 781–786.
16. Gadjiev A.D., Orhan C., Some approximation properties via statistical convergence, Rocky Mountain J. Math. (2002), 32, 129–138.
17. Gauchman H., Integral inequalities in q calculus. Comput Math. Appl. (2004), 47, 281–300.
18. Ispir N., On modi…ed Baskakov operators on weighted spaces, Turk. J. Math. (2001), 26, –365.
19. Lorentz G.G., Bernstein Polynomials, Toronto, Canada, University of Toronto Press, 1953.
20. Marinkovi S., Rajkovi P., Stankovi M., The inequalities for some types of q integrals, Com- put. Math. Appl. (2008), 56, 2490–2498.
21. Phillips G.M., Bernstein polynomials based on the q integers. Ann. Numer. Math. (1997), , 511–518.
22. Steinhaus H., Sur la convergence ordinaire et la convergence asymptotique, Colloq Math. (1951) 2, 73–74.
23. Current address : A. Karaisa, Department of Mathematics–Computer Sciences, Necmettin Erbakan University, 42090 Meram, Konya, Turkey
24. E-mail address : [e-mail], [e-mail] Current address : A. Aral, Department of Mathematics, Kırıkkale University,71450 Yah¸sihan, Kırıkkale, Turkey
25. E-mail address : [e-mail]