Approximation theorems for Kantorovich type Lupaş-Stancu operators based on \(q\)-integers

Main Article Content

Sevilay Kirci Serenbay
Özge Dalmanoğlu


In this paper, we introduce a Kantorovich generalization of q-Stancu-Lupa¸s operators and investigate their approximation properties. The rate of convergence of these operators are obtained by means of modulus of continuity, functions of Lipschitz class and Peetre's K-functional. We also investigate the convergence of the operators in the statistical sense and give a numerical example in order to estimate the error in the approximation.

Article Details

How to Cite
Serenbay, S., & Dalmanoğlu, Özge. (2017). Approximation theorems for Kantorovich type Lupaş-Stancu operators based on \(q\)-integers. J. Numer. Anal. Approx. Theory, 46(1), 78-92. Retrieved from


[1] P.N. Agrawal, N. Ispir and A. Kajla Approximation Properties of Lupa¸s-Kantorovich operators based on Polya distribution, Rend. Circ. Mat.Palermo (2016) 65: 185-208.

[2] A. Aral, V. Gupta, R. P. Agarwal, Applications of q-Calculus in Operator Theory, Springer, Berlin, 2013.

[3] O. Dogru and K. Kanat, Statistical Approximation Properties of King-type modification of Lupa¸s Operators, Computers and Mathematics with Applications, 64 (2012) 511-517.

[4] O. Dogru and K. Kanat, On statistical approximation properties of the Kantorovich type Lupa¸s operators, Mathematical and Computer Modelling, 55, 3–4, (2012), 1621

[5] O. Dogru, G. ·Içöz and K. Kanat, On the Rates of Convergence of the q-Lupa¸s-Stancu Operators, Filomat 30:5 (2016), 1151-1160.

[6] H. Fast, Sur la convergence statistique, Colloq. Math Studia Mathematica, 2 (1951), 241-244

[7] A.D. Gadjiev and C. Orhan, Some approximation properties via statistical convergence, Rocky Mountain J. of Math., 32 (2002), 129-138

[8] V. Kac and P. Cheung, Quantum Calculus, Springer-Verlag, New York Berlin Heidelberg, 1953.

[9] A. Lupa¸s, A q-analogue of the Bernstein operator, University of Cluj-Napoca, Seminar on numerical and statistical calculus, 1987.

[10] N.I. Mahmudov and P. Sabancgil, Approximation Theorems for q-Bernstein-Kantorovich Operators, Filomat 27:4 (2013) 721-730.

[11] N.I. Mahmudov and P. Sabancgil, Voronovskaja type theoremfor the Lupa¸s q-analogue of the Bernstein operators, Math. Commun. 17, (2012) 83-91.

[12] Niven I. , Zuckerman H.S. and Montgomery H., An Introduction to the Theory of Numbers, 5th edition, Wiley, New York, 1991

[13] A.M. Acu, D. Barbasu, D.F. Sofonea, Note on a q-Analogue of Stancu-Kantorovich Operators, Miskolc Mathematical Notes, 16 (2015) 1, 3-15.

[14] A.M. Acu, Stancu-Schurer-Kantorovich operators based on q-integers, Applied Mathematics and Computation, 259 (2015) 896-907.

[15] S. Ostrovska, On the Lupa¸s q-Analogue of the Bernstein Operator, Rocky Mountain Journal of Mathematics, 36 5 (2006), 1615-1629.

[16] M.A. Özarslan and T. Vedi, q-Bernstein-Schurer-Kantorovich Operators, Journal of Inequalities and Applications, (2013) 444.