Approximation of Modified Jakimovski-Leviatan-Beta Type Operators

Abstract

In the present paper, we define Jakimovski-Leviatan type modified operators. We study some approximation results for these operators. We also determine the order of convergence in terms of modulus of continuity, Lipschitz functions, Peetre's K-functional, second order modulus of continuity and weighted modulus of continuity.

Keywords

• Jakimovski-Leviatan operators,Korovkin's theorem,Modulus of continuity,Rate of convergence,$ K $-functional,Weighted space

References

1. [1] T. Acar, Asymptotic formulas for generalized Szász-Mirakyan operators, Appl. Math. Comput., 263 (2015) 223– 239.
2. [2] A. Ciupa, A class of integral Favard-Szász type operators, Stud. Univ. Babe¸s-Bolyai, Math., 40 (1995) 39–47.
3. [3] W. A. Al-Salam, q-Appell polynomials. Ann. Mat. Pura Appl., 4 (1967) 31–45.
4. [4] P. Appell, Une classe de polynômes, Ann. Sci. École Norm. Sup., 9 (1880) 119–144.
5. [5] A. Aral, T. Acar, Weighted approximation by new Bernstein-Chlodowsky- Gadjiev operators, Filomat, 27 (2013) 371–380.
6. [6] C. Atakut, I. Büyükyazici, Approximation by modified integral type Jakimovski-Leviatan operators, Filomat, 30 (2016) 29–39.
7. [7] İ. Büyükyazıcı, H. Tanberkan, S. Serenbay, C. Atakut, Approximation by Chlodowsky type Jakimovski-Leviatan operators, Jour. Comput. Appl. Math., 259 (2014) 153–163.
8. [8] J. Choi, H.M. Srivastava, q-Extensions of a multivariable and multiparameter generalization of the Gottlieb polynomials in several variables, Tokyo J. Math., 37 (2014) 111–125.

9. [9] A. D. Gadzhiev, A problem on the convergence of a sequence of positive linear operators on unbounded sets, and theorems that are analogous to P. P. Korovkin’s theorem. Dokl. Akad. Nauk SSSR (Russian), 218 (1974) 1001–1004.
10. [10] A. D. Gadzhiev, Weighted approximation of continuous functions by positive linear operators on the whole real axis, Izv. Akad. Nauk Azerbaijan. SSR Ser. Fiz.-Tehn. Mat. Nauk (Russian), 5 (1975) 41–45.
11. [11] P. Gupta, P. N. Agarwal, Jakimovski-Leviatan operators of Durrmeyer type involving involving Appell polynomials, Turk J. Math., 42 (2018) 1457–1470.
12. [12] F.H. Jackson, On q-definite integrals, Quart. J. Pure Appl. Math., 41(15) (1910) 193–203.
13. [13] A. Jakimovski, D. Leviatan, Generalized Szasz operators for the approximation in the infinite interval. Mathematica (Cluj), 11 (34) (1969) 97-103.
14. [14] V. Kac., A. De Sole, On integral representations of q-gamma and q-beta functions, Rend. Mat. Acc. Lincei, 9 (200) 11–29.
15. [15] M. E. Keleshteri, N.I. Mahmudov, A study on q-Appell polynomials from determinantal point of view, Appl. Math. Comp., 260 (2015) 351–369.
16. [16] P. P. Korovkin, Linear Operators And Approximation Theory, Hindustan Publ. Co. Delhi, 1960.
17. [17] G. V. Milovanovic, M. Mursaleen., M. Nasiruzzaman, Modified Stancu type Dunkl generalization of Szász- Kantorovich operators, Rev. R. Acad. Cienc. Exactas Fis. Nat. Ser. A Math. RACSAM, 112(1) (2018) 135–151.
18. [18] M. Mursaleen, K.J. Ansari, M. Nasiruzzaman, Approximation by q-analogue of Jakimovski-Leviatan operators involving q-Appell polynomials, Iran. J. Sci. Technol. Trans. Sci. 41 (2017) 891–900.
19. [19] A. Wafi, N. Rao, D. Rai, Appproximation properties by generalized-Baskakov-Kantrovich-Stancu type operators, Appl. Math. Inform. Sci. Lett., 4 (2016) 111–118.
20. [20] B. Wood, Generalized Szász operators for the approximation in the complex domain, SIAM J. Appl. Math., 17 (1969) 790-801.