## Rate of convergence of Chlodowsky type Durrmeyer Jakimovski-Leviatan operators.(English)Zbl 1371.41007

Summary: In this paper, we determine the rate of pointwise convergence of the Chlodowsky type Durrmeyer Jakimovski-Leviatan operators $$L^\ast_n(f,x)$$ for functions of bounded variation. We use some methods and techniques of probability theory to prove our main result.

### MSC:

 41A10 Approximation by polynomials 41A25 Rate of convergence, degree of approximation 41A36 Approximation by positive operators
Full Text:

### References:

 [1] R. Bojanic, M. Vuilleunier, On the rate of convergence of Fourier Legendre series of functions of bounded variation, J. Approx. Theory, 31 (1981) 67-79. [2] İ. Büyükyazιcι, H. Tanberkan, S. Kirci Serenbay, Ç. Atakut, Approximation by Chlodowsky type Jakimovski-Leviatan operators, J. Comp. App. Math., 259 (2014) 153-163. · Zbl 1291.41018 [3] F. Cheng, On the rate of convergence of Bernstein polynomials of functions of bounded variation, J. Approx. Theory, 39 (1983) 259-274. · Zbl 0533.41020 [4] V. Gupta, Rate of convergence on Baskakov-Beta-Bézier operators for bounded variation functions, Int. J. Math. Math. Sci., 32(8) (2002) 471-479. · Zbl 1034.41011 [5] A. Jakimovski, D. Leviatan, Generalized Szász operators for the approximation in the infinite interval, Mathematica (Cluj), 11(34) (1969) 97-103. · Zbl 0188.36801 [6] A. Karaisa, Approximation by Durrmeyer type Jakimovski-Leviatan operators, Math. Meth. Appl. Sci., 39 (2016): 2401-2410. doi: 10.1002/mma.3650. · Zbl 1342.41029 [7] M. Mursaleen, Khursheed J. Ansari, On Chlodowsky variant of Szász operators by Brenke type polynomials, Appl. Math. Comput., 271 (2015) 991-1003. [8] M. Mursaleen, K.J. Ansari, Asif Khan, Approximation by a Kantorovich type q-Bernstein-Stancu operators, Complex Analysis and Operator Theory, 11(1) (2017) 85-107. · Zbl 1349.39053 [8] M. Mursaleen, Md. Nasiruzzaman, A. Alotaibi, On modified Dunkl generalization of Szsz operators via q-calculus, Jou. Ineq. Appl., (2017) 2017: 38. · Zbl 1357.41025 [9] M. Mursaleen, S. Rahman and A. Alotaibi, Dunkl generalization of q-Szász-Mirakjan-Kantorovich operators which preserve some test functions, Jou. Ineq. Appl., (2016) 2016: 317. · Zbl 1351.41012 [10] H. M. Srivastava, Z. Finta, V. Gupta, Direct results for a certain family of summation-integral type operators, Appl. Math. Comput., 190 (2007) 449-457. · Zbl 1124.41020 [11] H. M. Srivastava, V. Gupta, A certain family of summation-integral type operators, Math. Comput. Model., 37 (2003) 1307-1315. · Zbl 1058.41015 [12] H. M. Srivastava, V. Gupta, Rate of convergence for the Bézier variant of the Bleimann-Butzer-Hahn operators, Appl. Math. Lett., 18 (2005) 849-857. · Zbl 1084.41010 [13] H. M. Srivastava, M. Mursaleen, Abdullah M. Alotaibi, Md. Nasiruzzaman, A. A. H. Al-Abied, Some approximation results involving the q-Szász-Mirakjan-Kantorovich type operators via Dunkl’s generalization, Math. Meth. Appl. Sci., DOI:10.1002/mma.4397. · Zbl 1384.41015 [14] H. M. Srivastava, X.-M. Zeng, Approximation by means of the Szász-B_ezier integral operators, Internat. J. Pure Appl. Math., 14 (2004) 283-294. [15] X.M. Zeng, W. Chen, On the rate of convergence of the generalized Durrmeyer type operators for functions of bounded variation, J. Approx. Theory, 102 (2000) 1-12. · Zbl 0956.41013 [16] X.M. Zeng, J. N. Zhao, Exact bounds for some basis function of approximation operators, J. Inequal. Appl., 6(2001) 563-575. · Zbl 0991.41016
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.