Approximation properties of (p, q)-Bernstein type operators

Zoltán Finta

doi:10.1515/ausm-2016-0014

Abstract

We introduce a new generalization of the q-Bernstein operators involving (p, q)-integers, and we establish some direct approximation results. Further, we define the limit (p, q)-Bernstein operator, and we obtain its estimation for the rate of convergence. Finally, we introduce the (p, q)-Kantorovich type operators, and we give a quantitative estimation.

References

[1] A. Aral, V. Gupta, R. P. Agarwal, Applications of q-Calculus in Operator Theory, Springer, New York, 2012.10.1007/978-1-4614-6946-9
Search in Google Scholar Back to article
[2] H. Gauchman, Integral inequalities in q-calculus, Comput. Math. Appl., 47 (2004), 281–300.10.1016/S0898-1221(04)90025-9
Search in Google Scholar Back to article
[3] V. Gupta, R. P. Agarwal, Convergence Estimates in Approximation Theory, Springer, New York, 2014.10.1007/978-3-319-02765-4
Search in Google Scholar Back to article
[4] A. Il’inskii, S. Ostrovska, Convergence of generalized Bernstein polynomials, J. Approx. Theory., 116 (1) (2002), 100–112.10.1006/jath.2001.3657
Search in Google Scholar Back to article
[5] V. Kac, P. Cheung, Quantum Calculus, Springer, New York, 2002.10.1007/978-1-4613-0071-7
Search in Google Scholar Back to article
[6] K. Khan, D. K. Lobiyal, Bézier curves based on Lupaş (p, q)-analogue of Bernstein polynomials in CAGD, arXiv:1505.01810[cs.GR].
Search in Google Scholar Back to article
[7] A. Lupaş, A q-analogue of the Bernstein operator, Seminar on Numerical and Statistical Calculus, 9 (1987), 85–92.
Search in Google Scholar Back to article
[8] S. Marinković, P. Rajković, M. Stanković, The inequalities for some type of q-integrals, Comput. Math. Appl., 56 (2008), 2490–2498.10.1016/j.camwa.2008.05.035
Search in Google Scholar Back to article
[9] M. Mursaleen, K. J. Ansari, A. Khan, On (p, q)-analogue of Bernstein operators, Appl. Math. Comput., 266 (2015), 874–882.
Search in Google Scholar Back to article
[10] M. Mursaleen, K. J. Ansari, A. Khan, Some approximation results for Bernstein-Kantorovich operators based on (p, q)-calculus, arXiv:1504.05887v4[math.CA] 15 Jan 2016.
Search in Google Scholar Back to article
[11] S. Ostrovska, The first decade of the q-Bernstein polynomials: results and perspectives, J. Math. Anal. Approx. Theory, 2 (1) (2007), 35–51.
Search in Google Scholar Back to article
[12] G. M. Phillips, Bernstein polynomials based on the q-integers, Ann. Numer. Math., 4 (1997), 511–518.
Search in Google Scholar Back to article
[13] V. Sahai, S. Yadav, Representations of two parameter quantum algebras and p, q-special functions, J. Math. Anal. Appl., 335 (2007), 268–279.10.1016/j.jmaa.2007.01.072
Search in Google Scholar Back to article
[14] P. N. Sadjang, On the (p, q)-Gamma and the (p, q)-Beta functions, arXiv:1506.07394v1. 22 Jun 2015.
Search in Google Scholar Back to article
[15] P. N. Sadjang, On the fundamental theorem of (p, q)-calculus and some (p, q)-Taylor formulas, arXiv:1309.3934[math.QA] 22 Aug 2013.
Search in Google Scholar Back to article
[16] H. Wang, F. Meng, The rate of convergence of q-Bernstein polynomials for 0 < q < 1, J. Approx. Theory, 136 (2005), 151–158.10.1016/j.jat.2005.07.001
Search in Google Scholar Back to article

Approximation properties of (p, q)-Bernstein type operators

Abstract

Paradigm

My account