Generalized Szász-Kantorovich Type Operators


  • Arun Kajla Department of Mathematics Central University of Haryana, Haryana 123031
  • Serkan Araci Department of Economics, Faculty of Economics, Administrative and Social Sciences, Hasan Kalyoncu University, TR-27410 Gaziantep
  • Meenu Goyal School of Mathematics, Thapar Institute of Engineering and Technology, Patiala 147004, Punjab
  • Mehmet Acikgoz Department of Mathematics, Faculty of Science and Arts, Gaziantep University, TR-27310 Gaziantep



Positive approximation process, Rate of convergence, Modulus of continuity, Steklov mean


In this note, we present Kantorovich modification of the operators introduced by V. Miheşan [Creative Math. Inf. 17 (2008), 466 – 472]. First, we derive some indispensable auxiliary results in the second section. We present a quantitative Voronovskaja type theorem, local approximation theorem by means of second order modulus of continuity and weighted approximation for these operators. Furthermore, we show the rate of convergence of these operators to certain functions with the help of the illustrations using Maple algorithms.


Download data is not yet available.


T. Acar, A. Aral, D. Cárdenas-Morales and P. Garrancho, Szász-Mirakyan type operators which fix exponentials, Results Math. 72(3) (2017), 1393 – 1404, DOI: 10.1007/s00025-017-0665-9.

T. Acar, A. Aral and S. A. Mohiuddine, On Kantorovich modification of ((p,q))-Bernstein operators, Iran J. Sci. Technol. Trans. Sci. 42(3) (2018), 1459 – 1464, DOI: 10.1007/s40995-017-0154-8.

T. Acar, V. Gupta and A. Aral, Rate of convergence for generalized Szász operators, Bull. Math. Sci. 1(1) (2011), 99 – 113, DOI: 10.1007/s13373-011-0005-4.

P. N. Agrawal, M. Goyal and A. Kajla, (q)-Bernstein-Schurer-Kantorovich type operators, Boll. Unione Mat. Ital 8 (2015), 169, DOI: 10.1007/s40574-015-0034-0.

P. N. Agrawal, V. Gupta, A. S. Kumar and A. Kajla, Generalized Baskakov-Szász type operators, Appl. Math. Comput. 236 (2014), 311 – 324, DOI: 10.1016/j.amc.2014.03.084.

A. Aral, A generalization of Szász-Mirakyan operators based on (q)-integers, Math. Comput. Modelling 47(9-10) (2008), 1052 – 1062, DOI: 10.1016/j.mcm.2007.06.018.

A. Aral, D. Inoan and I. Raşa, On the generalized Szász-Mirakyan operators, Results Math. 65(3-4) (2014), 441 – 452, DOI: 10.1007/s00025-013-0356-0.

í‡. Atakut and I. Büyükyazici, Approximation by Kantorovich-Szász type operators based on Brenke type polynomials, Numer. Funct. Anal. Optim. 37 (2016), 1488 – 1502, DOI: 10.1080/01630563.2016.1216447.

C. Bardaro, G. Vinti, P. L. Butzer and R. L. Stens, Kantorovich-type generalized sampling series in the setting of Orlicz spaces, Sampl. Theory Signal Image Process. 6 (2007), 29 – 52.

V. A. Baskakov, A sequence of linear positive operators in the space of continuous functions, Dokl. Acad. Nauk. SSSR 113 (1957), 249 – 251.

S. N. Bernstein, Demonstration du theoreme de weierstrass fondee sur le calcul de probabilities, Commun. Soc. Math. Kharkow 13(2) 1-2, 1912 – 1913.

P. L. Butzer, On the extensions of Bernstein polynomials to the infinite interval, Proc. Amer. Math. Soc. 5 (1954), 547 – 553.

A. Ciupa, On a generalized Favard-Szász type operator, Research Seminar on Numerical and Statistical Calculus, Univ. Babe¸s Bolyai Cluj-Napoca, preprint 1 (1994), 33 – 38.

O. Duman, M. A. í–zarslan and B. D. Vecchia, Modified Szász-Mirakjan-Kantorovich operators preserving linear, Turk. J. Math. 33 (2009), 151 – 158.

Z. Finta, N. K. Govil and V. Gupta, Some results on modified Szász-Mirakjan operators, J. Math. Anal. Appl. 327 (2007), 1284 – 1296, DOI: 10.3906/mat-0801-2.

A. D. Gadjiev, On P.P. Korovkin type theorems, Math. Zametki 20(5) (1976), 781 – 786, DOI: 10.1016/j.jmaa.2006.04.070.

V. Gupta and R. P. Agarwal, Convergence Estimates in Approximation Theory, Springer (2014), DOI: 10.1007/978-3-319-02765-4.

E. Ibikli and E. A. Gadjieva, The order of approximation of some unbounded function by the sequences of positive linear operators, Turkish J. Math. 19(3) (1995), 331 – 337.

A. Kajla and P. N. Agrawal, Szász-Durrmeyer type operators based on Charlier polynomials, Appl. Math. Comput. 268 (2015), 1001 – 1014, DOI: 10.1016/j.amc.2015.06.126.

A. Kajla, A. M. Acu and P. N. Agrawal, Baskakov-Szász type operators based on inverse Pólya-Eggenberger distribution, Ann. Funct. Anal. 8 (2017), 106 – 123, DOI: 10.1215/20088752-3764507.

A. Kajla and P. N. Agrawal, Approximation properties of Szász type operators based on Charlier polynomials, Turk. J. Math. 39 (2015), 990 – 1003, DOI: 10.3906/mat-1502-80.

A. Lupaş, The approximation by means of some linear positive operators, in Approximation Theory, Proceedings of the International Dortmund Meeting on Approximation Theory, Berlin, Germany, 1995, (M.W. Müller, M. Felten and D.H. Mache Eds.), Akademic Verlag, Berlin (1995), 201 – 229.

V. Miheşan, Gamma approximating operators, Creative Math. Inf. 17 (2008), 466 – 472.

S. M. Mazhar and V. Totik, Approximation by modified Szász operators, Acta Sci. Math. 49 (1985), 257 – 269.

M. Mursaleen, A. Alotaibi and K. J. Ansari, On a Kantorovich variant of Szász-Mirakjan operators, J. Funct. Spaces 2016 (2016), Article ID 1035253, 9 pages, DOI: 10.1155/2016/1035253.

M. A. í–zarslan and H. Aktuğlu, Local approximation properties for certain King type operators, Filomat 27(1) (2013), 173 – 181, DOI: 10.2298/FIL1301173O.

O. T. Pop, D. Miclćƒuş and D. Bćƒrbosu, The Voronovskaja type theorem for a general class of Szász-Mirakjan operators, Miskolc Math. Notes 14 (2013), 219 – 231, DOI: 10.18514/MMN.2013.374.

S. Sucu, Dunkl analogue of Szász operators, Appl. Math. Comput. 244 (2014), 42 – 48, DOI: 10.1016/j.amc.2014.06.088.

O. Szász, Generalization of S. Bernstein's polynomials to the infinite interval, J. Res. Nat. Bur. Standards 45 (1950), 239 – 245.

S. Varma and F. Taşdelen, Szász type operators involving Charlier polynomials, Math. Comput. Modelling 56 (2012), 118 – 122, DOI: 10.1016/j.mcm.2011.12.017.

I. Yüksel and N. Ispir, Weighted approximation by a certain family of summation integral-type operators, Comput. Math. Appl. 52(10-11) (2006), 1463 – 1470, DOI: 10.1016/j.camwa.2006.08.031.




How to Cite

Kajla, A., Araci, S., Goyal, M., & Acikgoz, M. (2019). Generalized Szász-Kantorovich Type Operators. Communications in Mathematics and Applications, 10(3), 403–413.



Research Article