Article http://dx.doi.org/10.26855/jamc.2021.12.016

Weighted Approximation Properties of New (p, q)—Analogue of Balazs Szabados Operators


Hayatem Hamal1,2,*, Pembe Sabancıgil1,2

1Department of Mathematics, Faculty of Education Janzour, Tripoli University, Tripoli, Libya.

2Department of Mathematics, Eastern Mediterranean University, Gazimagusa, North Cyprus.

*Corresponding author: Hayatem Hamal

Published: December 30,2021


Korovkin-type theorems provide simple and useful tools for finding out whether a given sequence of positive linear operators, acting on some function space is an approximation processor, equivalently, converges strongly to the identity operator. These theorems exhibit a variety of test subsets of functions which guarantee that the approximation property holds on the whole space provided it holds on them. These kinds of results are called “Korovkin-type theorems” which refers to P.P. Korovkin who in 1953 discovered such a property for the functions 1, X and X2   in the space C([0,1]). After this discovery, several mathematicians have undertaken the program of extending Korovkin’s theorems in many ways and to several settings. Such developments delineated a theory which is nowadays referred to as Korovkin-type approximation theory. In this paper, we study weighted approximation properties of new (p, q) - analogue of the Balázs-Szabados operators by using the weighted modulus of continuity and we give a Korovkin type theorem for weighted approximation.


[1] Balázs, K. (1975). Approximation by Bernstein type rational function. Acta Math. Acad. Sci. Hungar, Vol. 26,   No. 1-2, 123-134.

[2] Balázs, K. and Szabados, J. (1982). Approximation by Bernstein type rational function II. Acta Math. Acad. Sci. Hungar, Vol. 40, No. 3-4, 331-337.

[3] Hamal, H. and Sabancigil, P. (2020). Some Approximation properties of new Kantorovich type q-analogue of Balazs-Szabados Operators. J. Inequal. Appl, Vol: 159, No: 1-16.

[4] Doğru, O. (2006). On Statistical Approximation Properties of Stancu type bivariate generalization of  Balázs-Szabados operators. Proceedings. Int. Conf. on Numer. Anal. and Approx. Theory Cluj-Napoca, Romania. 179-194.

[5] Özkan, E. Y. (2019). Approximation Properties of Kantorovich type Balázs-Szabados operators. Demonstr. Math., 52, 10-19.

[6] Mahmudov, N. I. (2016). Approximation Properties of the Balázs-Szabados Complex Operators in the case. Comput. Methods Funct. Theory. Vol: 16, 567-583.

[7] İspir, N., Özkan, E. Y. (2013). Approximation Properties of Complex Balázs-Szabados Operators in Compact Disks. J. Inequal and Appl., 361.

[8] Mursaleen, M., Ansari, K. J., and Khan, A. (2015). On analogue of Bernstein operators, Appl. Math. Comput, (266),874-882. [Erratum: Appl. Math. Comput. 278, 70-71 (2016)].

[9] Mursaleen, M., Sarsenbi, A. M., Khan, T. (2016). On analogue of two parametric Stancu-Beta operators. J. Inequal. Appl, Artical ID 190.

[10] Mursaleen, M., Khan, A. (2014). Statistical approximation for new positive linear operators of lagrange type. Appl. Math. Comput., 232, 548-558. 

[11] Mursaleen, M., Ansari, K. J., and Khan, A. (2016). Approximation by Lorentz polynomials on a compact disk, Comlex Anal.Oper. Theory, 10(8), 1725-1740.

[12] Mursaleen, M., Nasiruzzaman, M., Khan, A., and Ansari, K. J. (2016). Some approximation results on Bleimann-Butzer-Hahn operators defined by integers, Filomat, (30) (3), 639-648.

[13] Mursaleen, M., Ansari, K. J., and Khan, A. (2016). Some approximation results for Bernstein-Kantorovich operators based on calculus. U.P.B. Sci. Bull. Series A., (78) (4), 129-142. 

[14] Mursaleen, M., Ansari, K. J., and Khan, A. (2015). Some approximation results by analogue of Bernstein-Stancu operators. Appl. Math. Compt., (246), 392-402.

[15] Acar, T. (2016). Generalization of Szász-Mirakyan operators. Math. Methods Appl. Sci., 39 (15), 2685-2695.

[16] Acar, T., Aral, A., Mohiuddine, S. A. (2016). On Kantorovich modification of Baskakov operators. J. Inequal. Appl., (98).

[17] Özkan, E. Y., Ispir N. (2018). Approximation by Analogue of Balázs-Szabados Operators. Filomat, 32(6), 2257-2271.

[18] Hamal, H. and Sabancigil, P. (2021). Some Approximation Properties of new (p,q)-analogue of Balázs-Szabados Operators. J. Inequal. Appl., No: 162.

[19] Aral, A., Gupta,V., Agarwal, R. P. (2013). Applications of Calculus in Operator Theory. Springer, Chapters: 4, 5.

[20] Aral, A., Gupta, V. (2010). Convergence of analogue of Szasz-Beta operators. Appl. Math. Comput, Vol: 216, No: 2, 374-380.

[21] Gadzhiev, A. D. (1974). The convergence problem for a sequence of positive linear operators on unbounded sets, and theorems analogous to that of P.P Korovkin. Sov. Math. Dokl, Vol: 15, No: 5, 1433-1436.

[22] Gadzhiev, A. D. (1976). P. P. Korovkin type theorems, Mathem. Zametki, Vol: 20, No: 5, Engl. Transl. Math Notes, Vol: 20, No: 5-6, 995-998.

[23] Altomare, F. (2010). Korovkin-type Theorems and Approximation by Positive Linear Operators. Surv. Approx. Theory, 5, 92-164.

[24] Gadzhiev, A. D., Efendiyev, E., Ibikl. (2003). On Korovkin type theorem in the space of locally integrable functions. Czech. Math.J, Vol: 53, No: (128)(1), 45-53.

[25] Yuksel, I., Ispir, N. (2006). Weighted approximation by a certain family of summation integral-type operators. Comput. Math. Appl, Vol: 52, No: 10-11, 1463-1470.

[26] López-Moreno, A.-J. (2004). Weighted simultaneous approximation with Baskakov type operators. Acta Math. Hunger, Vol: 104, No: 1-2, 143-151.

How to cite this paper

Weighted Approximation Properties of New (p, q)—Analogue of Balazs Szabados Operators

How to cite this paper: Hayatem Hamal, Pembe Sabancıgil. (2021) Weighted Approximation Properties of New (p, q)—Analogue of Balazs Szabados OperatorsJournal of Applied Mathematics and Computation5(4), 373-381.

DOI: http://dx.doi.org/10.26855/jamc.2021.12.016