Home 9 Volume 9 Approximation by sequences of q-Szasz-operators generated by Dunkl exponential function
Open AccessArticle
Approximation by sequences of q-Szasz-operators generated by Dunkl exponential function

Department of Mathematics, Faculty of Science, University of Tabuk, PO Box-4279, Tabuk71491, Saudi Arabia.

Department of Mathematics, Aligarh Muslim University, Aligarh 202002, India.

* Corresponding Author
Annals of Communications in Mathematics 2023
, 6 (4),
Received: 29 September 2023 |
Accepted: 30 November 2023 |
Published: 31 December 2023


The main purpose of this article is to introduce a modification of q-Dunkl generalization of Szasz-operators. We obtain approximation results via well known Korovkin’s type theorem. Moreover, we obtain the order of approximation, rate of convergence, functions belonging to the Lipschitz class and some direct theorems.


Cite This Article

M. Mursaleen, Md. Nasiruzzaman*.
Approximation by sequences of q-Szasz-operators generated by Dunkl exponential function.

Annals of Communications in Mathematics,

6 (4):

[1] N. L. Braha, H. M. Srivastava, S. A. Mohiuddine, A Korovkin’s type approximation theorem for periodic functions via the statistical summability of the generalized de la Vallee Poussin mean, Appl. Math. Comput., ´ 228 (2014) 162-169.
[2] B. Cheikh, Y. Gaied, M. Zaghouani, A q-Dunkl-classical q-Hermite type polynomials, Georgian Math. J., 21(2) (2014) 125-137.
[3] A. Ciupa, A class of integral Favard-Szasz type operators, Stud. Univ. Babes¸-Bolyai, Math., 40(1) (1995) ´ 39-47.
[4] G. ˙Ic¸oz, B. C¸ ekim, Dunkl generalization of Sz ¯ asz operators via ´ q-calculus, Jour. Ineq. Appl., (2015), 2015: 284.
[5] A. Lupas¸, A q-analogue of the Bernstein operator, In Seminar on Numerical and Statistical Calculus, University of Cluj-Napoca, Cluj-Napoca, 9 (1987) 85-92.
[6] S. A. Mohiuddine, An application of almost convergence in approximation theorems, Appl. Math. Lett., 24(11) (2011) 1856-1860.
[7] M. Mursaleen, K.J. Ansari, Approximation of q-Stancu-Beta operators which preserve x 2 , Bull. Malaysian Math. Sci. Soc., DOI: 10.1007/s40840-015-0146-9.
[8] M. Mursaleen, A. Khan, Statistical approximation properties of modified q- Stancu-Beta operators, Bull. Malays. Math. Sci. Soc. (2), 36(3) (2013) 683-690.
[9] M. Mursaleen, A. Khan, Generalized q-Bernstein-Schurer operators and some approximation theorems, Jour. Function Spaces Appl., Volume (2013), Article ID 719834, 7 pages.
[10] M. Mursaleen, Faisal Khan, Asif Khan, Approximation properties for modified q-bernstein-kantorovich operators, Numerical Functional Analysis and Optimization, 36(9) (2015) 1178-1197.
[11] M. Mursaleen, Faisal Khan, Asif Khan, Approximation properties for King’s type modified q-BernsteinKantorovich operators, Math. Meth. Appl. Sci., 38 (2015) 5242-5252.
[12] M. Mursaleen, M. Nasiruzzaman, Abdullah Alotaibi, On Modified Dunkl generalization of Szasz operators via q-calculus, Journal of Inequalities and Applications (2017) 2017:38, DOI 10.1186/s13660-017-1311-5.
[13] M. Mursaleen, Taqseer Khan, Md. Nasiruzzaman, Approximating Properties of Generalized Dunkl Analogue of Szasz Operators, Appl. Math. Inf. Sci., 10(6) (2016) 1-8.
[14] G. V. Milovanovic, M. Mursaleen, Md. Nasiruzzaman, Modified Stancu type Dunkl generalization of Sz ´ asz- ´ Kantorovich operators, RACSAM DOI 10.1007/s13398-016-0369-0.
[15] M. Nasiruzzaman, A.F. Aljohani, Approximation by α-Bernstein-Schurer operators and shape preserving properties via q-analogue, Math. Meth. Appl. Sci., 46(2) (2023) 2354–2372
[16] M. Nasiruzzaman, H. M. Srivastava, S. A. Mohiuddine, Approximation Process Based on Parametric Generalization of Schurer-Kantorovich Operators and their Bivariate Form, Proc. Natl. Acad. Sci., India, Sect. A Phys. Sci. 92 (2022) 301–311
[17] M. Nasiruzzaman, A. Mukheimer, M. Mursaleen, Approximation results on Dunkl generalization of Phillips operators via q-calculus Advances in Difference Equations 2019 (2019), Article Id: 244
[18] M. Nasiruzzaman, N. Rao, A generalized Dunkl type modifications of Phillips-operators, J. Inequal. Appl. 2018 (2018), Article ID: 323
[19] M. Nasiruzzaman, M. Mursaleen, R. P. Agarwal, Modified Dunkl type generalization of Phillips operators and some approximation results, Dynamics of Continuous, Discrete and Impulsive Systems Series A: Mathematical Analysis, 27 (2020), 33–45
[20] M. Orkcu, O. Dogru, Weighted statistical approximation by Kantorovich type ˘ q-Szasz Mirakjan operators, Appl. Math. Comput., 217 (2011) 7913-7919.
[21] M. Orkcu, O. Dogru, ˘ q-Szasz-Mirakyan-Kantorovich type operators preserving some test functions, Appl. ´ Math. Lett., 24 (2011) 1588-1593.
[22] G.M. Phillips, Bernstein polynomials based on the q- integers, Ann. Numer. Math., 4 (1997) 511-518.
[23] S. Sucu, Dunkl analogue of Szasz operators, Appl. Math. Comput., 244 (2014) 42-48.
[24] O. Szasz, Generalization of S. Bernstein’s polynomials to the infinite interval, J. Res. Natl. Bur. Stand., 45 ´ (1950) 239-245.
[25] A. Wafi, N. Rao, D. Rai, Appproximation properties by generalized-Baskakov-Kantrovich-Stancu type operators, Appl.Math.Inh.Sci.Lett., 4(3) (2016) 111-118.
[26] A. Wafi, N. Rao, A generalization of Szasz-type operators which preserves constant and quadratic test ´ functions, Cogent Mathematics (2016), 3: 1227023.

  • Creative Commons License
  • Copyright (c) 2023 by the Author(s). Licensee Techno Sky Publications. This article is an open access article distributed under the terms and conditions of the Creative Commons Attribution (CC BY) license (https://creativecommons.org/licenses/by/4.0/).


    Submit a Comment

    Your email address will not be published. Required fields are marked *

    Preview PDF

    XML File