Abstract
In the present paper, we introduce Kantorovich modifications of (p, q)-Bernstein operators using a new (p, q) -integral. We first estimate the moments and central moments. We obtain uniform convergence of new operators, rate of convergence in terms of classical modulus of continuity and second order modulus of continuity. We also investigate the rate of convergence of new operators for functions belonging to Lipschitz class and finally, we give an upper bound for the error of approximation via modulus of continuity of the derivative of approximating function.
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References
Acar T (2015) Asymptotic formulas for generalized Szász–Mirakyan operators. Appl Math Comput 263:223–239
Acar T (2016) \((p, q)\)-generalization of Szász–Mirakyan operators. Math Methods Appl Sci 39(10):2685–2695
Acar T, Aral A (2015) On pointwise convergence of \(q\)-Bernstein operators and their \(q\)-derivatives. Numer Funct Anal Optim 36(3):287–304
Acar T, Aral A, Mohiudine SA (2016a) On Kantorovich modification of \((p, q)\)-Baskakov operators. J Inequal Appl 2016:98
Acar T, Aral A, Mohiudine SA (2016b) Approximation by bivariate \((p, q)\)-Bernstein–Kantorovich operators. Iran J Sci Technol Trans Sci. doi:10.1007/s40995-016-0045-4
Acar T, Agrawal PN, Kumar AS (2016c) On a modification of (p, q)-Szász–Mirakyan operators. Complex Anal Oper Theory. doi:10.1007/s11785-016-0613-9
Braha NL, Srivastava HM, Mohiuddine SA (2014) A Korovkin’s type approximation theorem for periodic functions via the statistical summability of the generalized de la Vallée Poussin mean. Appl Math Comput 228:162–169
Burban I (1995) Two-parameter deformation of the oscillator algebra and \((p, q)\) analog of two dimensional conformal field theory. Nonlinear Math Phys 2(3–4):384–391
Burban IM, Klimyk AU (1994) \(P, Q\) differentiation, \(P, Q\) integration and \(P, Q\) hypergeometric functions related to quantum groups. Integral Transform Spec Funct 2(1):15–36
Devore RA, Lorentz GG (1993) Constructive approximation. Springer, Berlin
Ditzian Z, Totik V (1987) Moduli of smoothness. Springer, New York
Hounkonnou MN, Désiré J, Kyemba B (2013) \({\cal{R}}(p, q)\)-calculus: differentiation and integration. SUT J Math 49(2):145–167
İlarslan HG, Acar T (2016) Approximation by bivariate (p, q)-Baskakov–Kantorovich operators. Georgian Math J. doi:10.1515/gmj-2016-0057
Jagannathan R, Rao KS (2005) Two-parameter quantum algebras, twin-basic numbers, and associated generalized hypergeometric series. In: Proceedings of the international conference on number theory and mathematical physics, pp 20–21
Mohiuddine SA (2011) An application of almost convergence in approximation theorems. Appl Math Lett 24:1856–1860
Mursaleen M, Ansari KJ, Khan A (2015a) On \((p,q)\)-analogue of Bernstein operators. Appl Math Comput 266:874–882 [Erratum: Appl Math Comput 278:70–71 (2016)]
Mursaleen M, Ansari KJ, Khan A (2015b) Some approximation results by \((p, q)\)-analogue of Bernstein–Stancu operators. Appl Math Comput 264:392–402
Mursaleen M, Nasiuzzaman Md, Nurgali A (2015) Some approximation results on Bernstein–Schurer operators defined by \((p, q)\)-integers. J Inequal Appl 2015:249
Mursaleen M, Khan F, Khan A (2016) Approximation by \((p, q)\)-Lorentz polynomials on a compact disk. Oper Theory Complex Anal. doi:10.1007/s11785-016-0553-4
Sadjang PN (2013) 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 335:268–279
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The authors gratefully acknowledge the financial support from King Abdulaziz University, Jeddah, Saudi Arabia.
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Acar, T., Aral, A. & Mohiuddine, S.A. On Kantorovich Modification of (p, q)-Bernstein Operators. Iran J Sci Technol Trans Sci 42, 1459–1464 (2018). https://doi.org/10.1007/s40995-017-0154-8
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DOI: https://doi.org/10.1007/s40995-017-0154-8