ON p; q -ANALOG OF STANCU-BETA OPERATORS AND THEIR APPROXIMATION PROPERTIES

Year 2018, Volume: 8 Issue: 1, 136 - 143, 01.06.2018

M. Mursaleen T. Khan

Abstract

In this paper we introduce the p; q -analogue of the Stancu-Beta operators and call them as the p; q -Stancu-Beta operators. We study approximation properties of these operators based on the Korovkin's approximation theorem and also study some direct theorems. Also, we study the Voronovskaja type estimate for these operators.

Keywords

p, q -calculus, Stancu-Beta operators, modulus of continuity, positive linear operators, Korovkin type approximation theorem, Lipschitz class of functions.

References

• Acar T., (2016), (p, q)-generalization of Sz´asz–Mirakyan operators, Math. Meth. Appl. Sci., 39 (10), pp. 2685-2695.
• Acar T., Aral A., Mohiuddine S. A., (2016), On Kantorovich modifications of (p, q)-Baskakov opera- tors, J. Inequal. Appl., 2016:98.
• Acar T., Aral A., Mohiuddine S. A., Approximation by bivariate (p, q)-Bernstein-Kantorovich opera- tors, Iran. J. Sci. Technol. Trans. A Sci., DOI: 10.1007/s40995-016-0045-4.
• Aral A., Gupta V., (2012), On the q-analogue of Stancu-Beta operators, Appl. Math. Letters, 25, pp. 67-71.
• Cai Q. B., Zhou G., (2016), On (p, q)-analogue of Kantorovich type Bernstein–Stancu–Schurer oper- ators, Appl. Math. Comput., 276, pp. 12-20.
• Devore R. A., Lorentz G. G., (1993), Constructive Approximation, Springer, Berlin.
• Gadzhiev A. D., (1976), Theorems of the type of P.P. Korovkin type theorems, Math. Zametki, 20 (5), pp. 781-786.
• Ilarslan H. G. I. and Acar T., Approximation by bivariate (p, q)-Baskakov–Kantorovich operators, Georgian Math. J., DOI: 10.1515/gmj-2016-0057.
• M.N. Hounkonnou, J. Dsir, B. Kyemba, (2013), R(p, q)-calculus: differentiation and integration, SUT Jour. Math., 49(2), pp. 145-167.
• Lupa¸s A., (1987), A q-analogue of the Bernstein operator, University of Cluj-Napoca, Seminar on Numerical and Statistical Calculus, 9, pp. 85-92.
• Mishra V. N., Pandey S., (2016), On Chlodowsky variant of (p, q) Katorovich-Stancu-Schurer opera- tors, Int. J. Anal. Appl., 11(1), pp. 28-39.
• Mursaleen M., Alotaibi A., Ansari K. J., (2016), On a Kantorovich variant of (p, q)-Sz´asz-Mirakjan operators, J. Funct. Spaces, 2016, Article ID 1035253, 9 pages.
• Mursaleen M., Ansari K. J., Khan A., (2016), On (p, q)-analogue of Bernstein operators, Appl. Math. Comput., 266 (2015), pp. 874-882 [Erratum: Appl. Math. Comput., 278, pp. 70-71].
• Mursaleen M., Ansari K. J., Khan A., (2015), Some approximation results by (p, q)-analogue of Bernstein-Stancu operators, Appl. Math. Comput., 264 (2015), pp. 392-402 [Corrigendum: Appl. Math. Comput, 269, pp. 744-746].
• Mursaleen M., Khan F., Khan A., (2016), Approximation by (p, q) -Lorentz polynomials on a compact disk, Complex Anal. Oper. Theory, 10(8), pp. 1725-1740.
• Mursaleen M., Nasiruzzaman Md., Khan A., Ansari K. J., (2016), Some approximation results on Bleimann-Butzer-Hahn operators defined by (p, q) -integers, Filomat, 30(3), pp. 639-648.
• Mursaleen M., Nasiuzzaman Md., Nurgali A., (2015), Some approximation results on Bernstein- Schurer operators defined by (p, q)-integers, J. Ineq. Appl., 2015:249.
• Mursaleen M. and Nasiruzzaman Md., (2017), Some approximation properties of bivariate Bleimann- Butzer-Hahn operators based on (p, q)-integers, Boll. Unione Mat. Ital., 10, pp. 271-289.
• Mursaleen M., Sarsenbi A. M., Khan T., (2016), On (p, q)-analogue of two parametric Stancu-Beta operators, J. Ineq. Appl., 2016:190.
• Phillips G. M., (1997), Bernstein polynomials based on the q -integers, The Heritage of P. L. Cheby- shev, Ann. Numer. Math., 4, pp. 511-518.
• Sadjang P. N., On the fundamental theorem of (p, q)-calculus and some (p, q)-Taylor formulas, arXiv: 1309.3934 [math.QA].
• Sahai V., Yadav S., (2007), Representations of two parameter quantum algebras and p, q-special functions, J. Math. Anal. Appl., 335, pp. 268-279.
• Sharma H., On Durrmeyer-type generalization of (p, q) -Bernstein operators, Arab. J. Math., DOI 10.1007/s40065-016-0152-2.
• Stancu D. D., (1995), On the beta approximating operators of second kind, Revue d’Analyse Num´erique et de Th´erie de l’Approximation, 24, pp. 231-239.