Dunkl Generalization of q-Parametric Szasz-Mirakjan Operators

Article Sidebar

PDF
Cite as:
M. Mursaleen, Md. Nasiruzzaman, A.A.H. Al-Abied, Dunkl Generalization of q-Parametric Szasz-Mirakjan Operators, Int. J. Anal. Appl., 13 (2) (2017), 206-215.

Main Article Content

M. Mursaleen, Md. Nasiruzzaman, A.A.H. Al-Abied

Abstract

In this paper, we construct q-parametric Szász-Mirakjan operators generated by the q-Dunkl generalization of the exponential function. We obtain Korovkin's type approximation theorem and compute convergence of these operators by using the modulus of continuity. Furthermore, we obtain the rate of convergence of these operators for functions belonging to the Lipschitz class.

Article Details

References

  1. M. Altinok, M. Kü ¸ cükaslan, A-statistical supremum-infimum and A-statistical convergence, Azerbaijan Journal of Mathematics, 4 (2) (2014), 31-42.
  2. A. Aral, V. Gupta, On q-analogue of Stancu-beta operators, Appl. Math. Letters, 25 (2012), 67-71.
  3. A. Aral, O. Dogru, Bleimann Butzer and Hahn operators based on q-integers, J. Ineq. Appl., 2007 (2007), Art. ID 79410.
  4. A. Aral, V. Gupta: The q-derivative and applications to q-Szász Mirakyan operators, Calcolo 43 (3) (2006), 151-170.
  5. S.N. Bernstein, Demonstration du theoreme de Weierstrass fondee sur le calcul des probabilit ´ es, Commun. Soc. Math. Kharkow, 2 (13) (1912), 1-2.
  6. B. Cheikh, Y. Gaied, M. Zaghouani, A q-Dunkl-classical q-Hermite type polynomials. Georgian Math. J., 21 (2) (2014), 125-137.
  7. A. Ciupa, A class of integral Favard-Szász type operators. Stud. Univ. Babe ¸ s-Bolyai, Math., 40 (1) (1995), 39-47.
  8. S. Ersan, O. Dogru, Statistical approximation properties of q-Bleimann, Butzer and Hahn operators, Math. Comput. Modell., 49 (2009), 1595-1606.
  9. A.D. Gadjiev, Simultaneous statistical approximation of analytic functions and their derivatives by k-positive linear operators, Azerbaijan Journal of Mathematics, 1 (1) (2011), 57-66.
  10. V. Gupta, C. Radu, Statistical approximation properties of q-Baskokov-Kantorovich operators, Cent. Eur. J. Math., 7 (4) (2009), 809-818.
  11. G. Ë™ I ¸ cöz, Bayram C ¸ekim, Dunkl generalization of Szász operators via q-calculus, Jour. Ineq. Appl., 2015 (2015), Art. ID 284.
  12. N. Ispir, Approximation by modified complex Szasz-Mirakjan operators, Azerbaijan Journal of Mathematics, 3 (2) (2013), 95-107.
  13. V. Kac, P. Cheung, Quantum Calculus. Springer-Verlag New York, 2002.
  14. V. Kac, P. Cheung, Quantum Calculu., Universitext, Springer-Verlag, New York, 2002.
  15. A. Lupa ¸ s, A q-analogue of the Bernstein operator, University of Cluj-Napoca, Seminar on numerical and statistical calculus, No. 9, 1987.
  16. N.I. Mahmudov, V. Gupta, On certain q-analogue of Szász Kantorovich operators, J. Appl. Math. Comput., 37 (2011) 407-419.
  17. M. Mursaleen, Asif Khan, H.M. Srivastava, K.S. Nisar, Operators constructed by means of q-Lagrange polynomials and A-statistical approximation, Appl. Math. Comput., 219 (2013), 6911-6918.
  18. M. Mursaleen, Asif Khan, Generalized q-Bernstein-Schurer operators and some approximation theorems, Jour. Function Spaces Appl., 2013 (2013), Article ID 719834, 7 pages.
  19. M. Mursaleen, Asif Khan, Statistical approximation properties of modified q-Stancu-Beta operators, Bull. Malaysian Math. Sci.Soc.(2), 36 (3) (2013), 683-690.
  20. M. Mursaleen, Faisal Khan and Asif Khan, Approximation properties for modified q-Bernstein-Kantorovich operators, Numer. Funct. Anal. Optim., 36(9) (2015), 1178-1197.
  21. M. Mursaleen, Faisal Khan and Asif Khan, Approximation properties for King's type modified q-Bernstein-Kantorovich operators, Math. Meth. Appl. Sci., 38 (2015), 5242-5252.
  22. M. Orkcü, O. Dogru, Weighted statistical approximation by Kantorovich type q-Szász Mirakjan operators, Appl. Math. Comput., 217 (2011), 7913-7919.
  23. G.M. Phillips, Bernstein polynomials based on the q- integers, The heritage of P.L. Chebyshev, A Festschrift in honor of the 70th-birthday of Professor T. J. Rivlin. Ann. Numer. Math., 4 (1997), 511-518.
  24. C. Radu, Statistical approximation properties of Kantorovich operators based on q-integers, Creat. Math. Inform., 17 (2) (2008), 75-84.
  25. M. Rosenblum, Generalized Hermite polynomials and the Bose-like oscillator calculus, Oper. Theory, Adv. Appl., 73 (1994), 369-396.
  26. S. Sucu, Dunkl analogue of Szász operators, Appl. Math. Comput., 244, (2014), 42-48.
  27. O. Szász, Generalization of S. Bernstein's polynomials to the infinite interval, J. Res. Natl. Bur. Stand., 45 (1950), 239-245.