Complex Analysis and Operator Theory

, Volume 12, Issue 7, pp 1519–1536 | Cite as

Szász-Durrmeyer Operators Based on Dunkl Analogue

  • Abdul Wafi
  • Nadeem RaoEmail author


In this article, we construct Sz\(\acute{a}\)sz-Durrmeyer type operators based on Dunkl analogue. We investigate several approximation results by these positive linear sequences, e.g. rate of convergence by means of classical modulus of continuity, uniform approximation using Korovkin type theorem on compact interval. Further, we discuss local approximations in terms of second order modulus of continuity, Peetre’s K-functional, Lipschitz type class and rth order Lipschitz-type maximal function. Weighted approximation and statistical approximation results are discussed in the last of this article.


Sz\(\acute{a}\)sz operators Linear positive operators Modulus of continuity Rate of convergence Dunkl analogue 

Mathematics Subject Classification

41A10 41A25 41A28 41A35 41A36 


  1. 1.
    Weierstrass, K.G.: Über die analyishe Darstellbar keit sogenonnter willkürlincher Funtionen einer reelen Veründerlichen. Sitz. Akad. Berl. 2, 633–693 (1885)Google Scholar
  2. 2.
    Berntein, S.N.: Démonstration du théorème de Weierstrass fondée sur le calcul de probabilités. Commun. Soc. Math. Kharkow. 13(2), 1–2 (1912)Google Scholar
  3. 3.
    Szász, O.: Generalization of S. Bernstein’s polynomials to the infinite interval. J. Res. Nat. Bur. Stand. 45, 239–245 (1950)MathSciNetCrossRefGoogle Scholar
  4. 4.
    Wafi, A., Rao, N., A generalization of Szász-type operators which preserves constant and quadratic test functions. Cogent. Math. 3, 1227023 (2016)Google Scholar
  5. 5.
    Wafi, A., Rao, N., Rai, D.: Approximation Properties by Generalized-BaskakovKantorovich-Stancu Type Operators. Appl. Math. Inf. Sci. Lett. 4(3), 111–118 (2016)CrossRefGoogle Scholar
  6. 6.
    Sucu, S., İbikli, E.: Approximation by Jakimovski–Leviatan type operators on a complex domain. Complex Anal. Oper. Theory 8(1), 177–188 (2014)MathSciNetCrossRefGoogle Scholar
  7. 7.
    Mursaleen, M.: Khursheed J. Ansari, On Chlodowsky variant of Szász operators by Brenke type polynomials. Appl. Math. Comput. 271, 991–1003 (2015)MathSciNetGoogle Scholar
  8. 8.
    Kajla, A., Agrawal, P.N.: Szász-Durrmeyer type operators based on Charlier polynomials. Appl. Math. Comput. 268, 1001–1014 (2015)MathSciNetzbMATHGoogle Scholar
  9. 9.
    Durrmeyer, J. L.: Une formule dinversion de la transformée de Laplace-applications à la théorie des moments. Thése de 3e cycle, Faculté des Sciences de l Universit de Paris, (1967)Google Scholar
  10. 10.
    Sucu, S., İbikli, E.: Rate of convergence for Szász type operators including Sheffer polynomials. Stud. Univ. Babeş-Bolyai Math. 58(1), 55–63 (2013)MathSciNetzbMATHGoogle Scholar
  11. 11.
    Varma, S., Taşdelen, F.: Szász type operators involving Charlier polynomials. Math. Comput. Modelling. 56(5–6), 118–122 (2012)MathSciNetCrossRefGoogle Scholar
  12. 12.
    Sucu, S.: Dunkl analogue of Szász operators. Appl. Math. Comput. 244, 42–48 (2014)MathSciNetzbMATHGoogle Scholar
  13. 13.
    Rosenblum, M.: Generalized Hermite polynomials and the Bose-like oscillator calculus. Oper. Theory Adv. Appl. 73, 369–396 (1994)MathSciNetzbMATHGoogle Scholar
  14. 14.
    Altomare, F., Campiti, M.: Korovkin-type approximation theory and its applications. Appendix A by Michael Pannenberg and Appendix B by Ferdinand Beckhoff. de Gruyter Studies in Mathematics, 17. Walter de Gruyter and Co, Berlin (1994)Google Scholar
  15. 15.
    Shisha, O., Bond, B.: The degree of convergence of linear positive operators. Proc. Nat. Acad. Sci. 60, 1196–1200 (1968)MathSciNetCrossRefGoogle Scholar
  16. 16.
    DeVore, R.A., Lorentz, G.G.: Constructive Approximation. Grudlehren der Mathematischen Wissenschaften [Fundamental principales of Mathematical Sciences]. Springer, Berlin (1993)Google Scholar
  17. 17.
    Özarslan, M.A., Aktuğlu, H.: Local approximation for certain King type operators. Filomat 27, 173–181 (2013)MathSciNetCrossRefGoogle Scholar
  18. 18.
    Lenze, B.: On Lipschitz type maximal functions and their smoothness spaces. Nederl Akad. Indag. Math. 50, 53–63 (1988)MathSciNetCrossRefGoogle Scholar
  19. 19.
    Gadziev, A.D.: Theorems of the type of P.P. Korovkin’s theorems. Mat. Zametki 20(5), 781–786 (1976). ((in Russian), Math. Notes 20 (5-6) 995-998(Engl. Trans.) (1976))MathSciNetGoogle Scholar
  20. 20.
    İbikli, E., Gadjieva, E.A.: The order of approximation of some unbounded functions by the sequence of positive linear operators. Turk J. Math. 19(3), 331–337 (1995)MathSciNetzbMATHGoogle Scholar
  21. 21.
    Gadjiev, A.D., Orhan, C.: Some approximation theorems via statistical convergence. Rocky Mt. J. Math. 32(1), 129–138 (2007)MathSciNetCrossRefGoogle Scholar
  22. 22.
    Duman, O., Orhan, C.: Statistical approximation by positive linear operators. Studia Math. 16(2), 187–197 (2004)MathSciNetCrossRefGoogle Scholar

Copyright information

© Springer International Publishing 2017

Authors and Affiliations

  1. 1.Department of MathematicsJamia Millia IslamiaNew DelhiIndia

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