The aim of the paper is to investigate a qanalogue of a general class of linear positive operators defined by Baskakov and developed by Mastroianni. Our results are the following: the moments of the operators are explicitly expressed with the help of new q-analogues of Stirling numbers, the rate of convergence is established in different function spaces by using both modulus of continuity and a certain weighted modulus of smoothness, the identification, as particular cases, of q-analogues for two classical sequences of positive approximation processes.


Octavian Agratini
Department of Mathematics, Babes-Bolyai University, Cluj-Napoca, Romania

Cristina Radu
Babes-Bolyai University, Cluj-Napoca, Romania


q-integers; Stirling numbers; linear positive operator; Bohman-Korovkin theorem; moduli of smoothness; rate of convergence.

Paper coordinates

O. Agratini, C. Radu, On q-Baskakov-Mastroianni operators, Rocky Mountain Journal of Mathematics, 42 (2012) no. 3, pp. 773-790, http://doi.org/10.1216/RMJ-2012-42-3-773


About this paper


Rocky Mountain Journal of  Mathematics

Publisher Name

Rocky Mountain Mathematics Consortium

Print ISSN


Online ISSN

google scholar link

1. M. Abramowitz and I.A. Stegun, eds., Handbook of mathematical functions with formulas, graphs and mathematical tables, National Bureau of Standards Appl. Math. 55, issued June, 1964.
2. F. Altomare and M. Campiti, Korovkin-type approximation theory and its applications, de Gruyter Stud. Math. 17, Walter de Gruyter, Berlin, 1994.
3. A. Aral, A generalization of Sz´asz-Mirakjan operators based on q-integers, Math. Comput. Model. 47 (2008), 1052 1062.
4. A. Aral and O. Dogru, Bleimann, Butzer and Hahn operators based on the q-integers, J. Inequal. Appl. 79410 (2007), 12 pages.
5. V.A. Baskakov, An example of a sequence of linear positive operators in the space of continuous functions, Dokl. Akad. Nauk. 113 (1957), 249 251.
6. M.M. Derriennic, Modified Bernstein polynomials and Jacobi polynomials in q-calculus, Rend. Circ. Mat. Palermo 76 (2005), 269 290.
7. O. Dogru and O. Duman, Statistical approximation of Meyer-Konig and Zeller operators based on q-integers, Publ. Math. Debrecen 68 (2006), 199 214.
8. T. Ernst, The history of q-calculus and a new method, U.U.D.M. 16, Uppsala, Departament of Mathematics, Uppsala University, 2000.
9. V. Kac and P. Cheung, Quantum calculus, Universitext, Springer-Verlag, New York, 2002.
10. A.-J. Lopez-Moreno, Weighted silmultaneous approximation with Baskakov type operators, Acta Math. Hungarica 104 (2004), 143 151.
11. A. Lupas, A q-analogue of the Bernstein operator, Seminar on Numerical and Statistical Calculus, University of Cluj-Napoca 9 (1987), 85 92.
12. N.I. Mahmudov and P. Sabancigil, q-Parametric Bleimann Butzer and Hahn operators, J. Inequal. Appl. 816367 (2008), 15 pages.
13. G. Mastroianni, Su un operatore lineare e positivo, Rend. Acc. Sc. Fis. Mat. Napoli 46 (1979), 161 176.
14. S. Ostrovska, On the Lupas q-analogue of the Bernstein operator, Rocky Mountain J. Math. 36 (2006), 1615 1629.
15. , On the improvement of analytic properties under the limit q-Bernstein operator, J. Approx. Theory 138 (2006), 37 53.
16. , The first decade of the q-Bernstein polynomials: Results and perspectives, J. Math. Anal. Approx. Theory 2 (2007), 35 51.
17. G.M. Phillips, Bernstein polynomials based on the q-integers, Ann. Numer. Math. 4 (1997), 511 518.
18. , Interpolation and approximation by polynomials, Springer-Verlag, New York, 2003.
19. C. Radu, On statistical approximation of a general class of positive linear operators extended in q-calculus, Appl. Math. Comput. 215 (2009), 2317 2325.
20. T. Trif, Meyer-Konig and Zeller operators based on the q-integers, Rev. Anal. Numer. Theor. Approx. 29 (2000), 221 229.
21. V.S. Videnskii, On some classes of q-parametric positive operators, Operator Theory Adv. Appl. 158 (2005), 213 222.

Related Posts