On a generalization of Bleimann, Butzer and Hahn operators based on q-integers


We propose a class of linear positive operators based on q-integers. These operators depend on a non-negative parameter and represent a generalization of the classical Bleimann, Butzer and Hahn operators. Approximation properties are presented and bounds of the error of approximation are established.


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


Linear positive operator; q-integers; Bleimann, Butzer and Hahn operator; Approximation process

Paper coordinates

O. Agratini, G. Nowak, On a generalization of Bleimann, Butzer and Hahn operators based on q-integers, Mathematical and Computer Modelling, 53 (2011) nos. 5-6, pp. 699-706, https://doi.org/10.1016/j.mcm.2010.10.006


About this paper


Mathematical and Computer Modelling

Publisher Name

Elsevier Ltd

Print ISSN


Online ISSN

google scholar link

[1] G. Bleimann, P.L. Butzer, L. Hahn, A Bernstein-type operator approximating continuous functions on the semi-axis, Indag. Math. 42 (1980) 255–262.
[2] R.A. Khan, A note on a Bernstein-type operators of Bleimann, Butzer and Hahn, in: P. Nevai, A. Pinkus (Eds.), Process in Approximation Theory, Academic Press, Boston, 1991, pp. 497–504.
[3] J.A. Adell, J. de la Cal, M. San Miguel, Inverse beta and generalized Bleimann–Butzer–Hahn operators, J. Approx. Theory 76 (1994) 54–64.
[4] O. Agratini, Approximation properties of a generalization of Bleimann, Butzer and Hahn operators, Math. Pannon. 9 (1998) 165–171.
[5] U. Abel, M. Ivan, Some identities for the operator of Bleimann, Butzer and Hahn involving divided differences, Calcolo 36 (1999) 143–160.
[6] H.M. Srivastava, V. Gupta, Rate of convergence for the Bézier variant of the Bleimann–Butzer–Hahn operators, Appl. Math. Lett. 18 (2005) 849–857.
[7] A. Aral, O. Doğru, Bleimann, Butzer and Hahn operators based on the q-integers, J. Inequal. Appl. (2007) Art. ID 79410, 12 pp.
[8] O. Doğru, V. Gupta, Monotonicity and the asymptotic estimate of Bleimann, Butzer and Hahn operators based on q-integers, Georgian Math. J. 12 (3) (2005) 415–422.
[9] S. Ersan, O. Doğru, Statistical approximation properties of q-Bleimann, Butzer and Hahn operators, Math. Comput. Modelling 49 (2009) 1595–1605.
[10] N.I. Mahmudov, P. Sabancigil, Some approximation properties of q-parametric BBH operators, J. Comput. Anal. Appl. 12 (1-A) (2010) 111–123.
[11] A. Lupaş, A q-analogue of the Bernstein operator, in: Seminar on Numerical and Statistical Calculus, University of Cluj-Napoca, vol. 9, 1987, pp. 85–92.
[12] G. Nowak, Approximation properties for generalized q-Bernstein polynomials, J. Math. Anal. Appl. 350 (2009) 50–55.
[13] O. Agratini, On a q-analogue of Stancu operators, Cent. Eur. J. Math. 8 (1) (2010) 191–198.
[14] V. Kac, P. Cheung, Quantum Calculus, in: Universitext, Springer, New York, 2002.
[15] G.M. Phillips, Interpolation and Approximation by Polynomials, in: CMS Books in Mathematics, vol. 14, Springer, Berlin, 2003.
[16] A.D. Gadjiev, P. Çakar, On uniform approximation by Bleimann, Butzer and Hahn operators on all positive semiaxis, Trans. Natl. Acad. Sci. Azerb. 19 (1999) 21–26.
[17] J. Bustamante, L. Morales de la Cruz, Positive linear operators and continuous functions on unbounded intervals, Jaen J. Approx. 1 (2) (2009) 145–173.
[18] F. Altomare, M. Campiti, Korovkin-Type Approximation Theory and its Applications, in: de Gruyter Studies in Mathematics, vol. 17, Walter de Gruyter, Berlin, 1994.
[19] P.M. Rajković, M.S. Stanković, S.D. Marinković, Mean value theorems in q-calculus, Mat. Vesnik 54 (2002) 171–178.

Related Posts