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On Kantorovich Modification of (pq)-Bernstein Operators

  • Tuncer Acar
  • Ali Aral
  • S. A. MohiuddineEmail author
Research Paper

Abstract

In the present paper, we introduce Kantorovich modifications of (pq)-Bernstein operators using a new (pq) -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.

Keywords

\((p, q)\)-Integers \((p, q)\)-Integral \((p, q)\)-Bernstein operators \((p, q)\)-Bernstein–Kantorovich operators Rate of convergence 

Notes

Acknowledgements

The authors gratefully acknowledge the financial support from King Abdulaziz University, Jeddah, Saudi Arabia.

References

  1. Acar T (2015) Asymptotic formulas for generalized Szász–Mirakyan operators. Appl Math Comput 263:223–239Google Scholar
  2. Acar T (2016) \((p, q)\)-generalization of Szász–Mirakyan operators. Math Methods Appl Sci 39(10):2685–2695MathSciNetCrossRefzbMATHGoogle Scholar
  3. Acar T, Aral A (2015) On pointwise convergence of \(q\)-Bernstein operators and their \(q\)-derivatives. Numer Funct Anal Optim 36(3):287–304MathSciNetCrossRefzbMATHGoogle Scholar
  4. Acar T, Aral A, Mohiudine SA (2016a) On Kantorovich modification of \((p, q)\)-Baskakov operators. J Inequal Appl 2016:98MathSciNetCrossRefzbMATHGoogle Scholar
  5. 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
  6. 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 zbMATHGoogle Scholar
  7. 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 228:162–169MathSciNetzbMATHGoogle Scholar
  8. 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):384–391MathSciNetCrossRefzbMATHGoogle Scholar
  9. 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):15–36MathSciNetCrossRefzbMATHGoogle Scholar
  10. Devore RA, Lorentz GG (1993) Constructive approximation. Springer, BerlinCrossRefzbMATHGoogle Scholar
  11. Ditzian Z, Totik V (1987) Moduli of smoothness. Springer, New YorkCrossRefzbMATHGoogle Scholar
  12. Hounkonnou MN, Désiré J, Kyemba B (2013) \({\cal{R}}(p, q)\)-calculus: differentiation and integration. SUT J Math 49(2):145–167MathSciNetzbMATHGoogle Scholar
  13. İlarslan HG, Acar T (2016) Approximation by bivariate (p, q)-Baskakov–Kantorovich operators. Georgian Math J. doi: 10.1515/gmj-2016-0057 Google Scholar
  14. 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 20–21Google Scholar
  15. Mohiuddine SA (2011) An application of almost convergence in approximation theorems. Appl Math Lett 24:1856–1860MathSciNetCrossRefzbMATHGoogle Scholar
  16. 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)]Google Scholar
  17. Mursaleen M, Ansari KJ, Khan A (2015b) Some approximation results by \((p, q)\)-analogue of Bernstein–Stancu operators. Appl Math Comput 264:392–402MathSciNetGoogle Scholar
  18. Mursaleen M, Nasiuzzaman Md, Nurgali A (2015) Some approximation results on Bernstein–Schurer operators defined by \((p, q)\)-integers. J Inequal Appl 2015:249MathSciNetCrossRefzbMATHGoogle Scholar
  19. 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
  20. Sadjang PN (2013) On the fundamental theorem of (p, q)-calculus and some (p, q)-Taylor formulas. arXiv:1309.3934 [math.QA]
  21. Sahai V, Yadav S (2007) Representations of two parameter quantum algebras and \(p, q\)-special functions. J Math Anal Appl 335:268–279MathSciNetCrossRefzbMATHGoogle Scholar

Copyright information

© Shiraz University 2017

Authors and Affiliations

  1. 1.Department of Mathematics, Faculty of Science and ArtsKirikkale UniversityKirikkaleTurkey
  2. 2.Operator Theory and Applications Research Group, Department of Mathematics, Faculty of ScienceKing Abdulaziz UniversityJeddahSaudi Arabia

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