Szász–Gamma Operators Based on Dunkl Analogue

  • Abdul Wafi
  • Nadeem RaoEmail author
Research Paper


In this article, we introduce Szász–Gamma operators based on Dunkl analogue. We discuss basic approximation results in terms of the classical Korovkin theorem and modulus of continuity. For our operators, we investigate local approximation results by means of Peetre’s K-functional, second order modulus of smoothness, the Lipschitz class and the Lipschitz maximal function. Next, we present weighted approximation theorems by means of a Korovkin type theorem and weighted modulus of continuity. Lastly, A-Statistical approximation result and rate of convergence for functions with derivative of bounded variation are also studied.


Dunkl analogue Durrmeyer operators Szász–Gamma operators Rate of convergence 

Mathematics Subject Classification

41A10 41A25 41A28 41A30 41A35 41A36 


  1. Altomare F, Campiti M (1994) Korovkin-type approximation theory and its applications, vol 17. Walter De Gruyter, Berlin (De Gruyter Studies in Mathematics, Appendix A By Michael Pannenberg and Appendix B By Fendinand Beckho)CrossRefzbMATHGoogle Scholar
  2. DeVore RA, Lorentz GG (1993) onstructive approximation. Grudlehren der Mathematischen Wissenschaften [Fundamental principales of mathematical sciences]. Springer, BerlinGoogle Scholar
  3. Duman O, Orhan C (2004) Statistical approximation by positive linear operators. Studi Math 16(2):187–197MathSciNetCrossRefzbMATHGoogle Scholar
  4. Gadziev AD (1976) Theorems of the type of P.P. Korovkin’s theorems. Mat Zametki 20(5):781–786 [(in Russian), Math. Notes 20(5–6) (1976) 995–998 (Engl. Trans.)]MathSciNetGoogle Scholar
  5. Gadjiev AD, Orhan C (2007) Some approximation theorems via statistical convergence. Rocky Mt J Math 32(1):129–138MathSciNetCrossRefzbMATHGoogle Scholar
  6. Ibikli E, Gadjieva EA (1995) The order of approximation of some unbounded functions by the sequence of positive linear operators. Turk J Math 19(3):331–337MathSciNetzbMATHGoogle Scholar
  7. İçöz G, Çekim B (2015a) Dunkl generalization of Szász operators via q-calculus. J Inequal Appl 2015:284CrossRefzbMATHGoogle Scholar
  8. İçöz G, Çekim B (2015b) Stancu-type generalization of Dunkl analogue of Szász-Kantorovich operators. Math Met Appl Sci.
  9. Kajla A, Agrawal PN (2015) Szász-Durrmeyer type operators based on Charlier polynomials. Appl Math Comput 268:1001–1014MathSciNetzbMATHGoogle Scholar
  10. Lenze B (1988) On Lipschitz type maximal functions and their smoothness spaces. Nederl Akad Indag Math 50:53–63MathSciNetCrossRefzbMATHGoogle Scholar
  11. Mazhar SM, Totik V (1985) Approximation by modified Szász operators. Acta Sci Math (Szeged) 49(1–4):257–269MathSciNetzbMATHGoogle Scholar
  12. Mursaleen M, Nasiruzzaman Md, Shrivastwa HM (2016) Approximation by bicomplex beta operators in compact \(({\cal{B}}) {\mathbb{C}}\)-disks. Math Met Appl Sci.
  13. Mursaleen M, Nasiruzzaman Md, Shrivastwa HM (2015) Approximation properties for modified q-Bernstein–Kantorovich operators. Numer Funct Anal Optim 36(5):1178–1197MathSciNetCrossRefGoogle Scholar
  14. Mursaleen M, Nasiruzzaman Md (2015) A Dunkl generalization of q-parametric Szász-Mirakjan operators. arXiv:1511.06628 [math.CA]
  15. Karaisa A, Karakoç F (2016) Stancu type generalization of Dunkl analogue of Szàsz Operators. Adv Appl Cliff Algebras 26:1235. CrossRefzbMATHGoogle Scholar
  16. Özarslan MA, Aktuğlu H (2013) Local approximation for certain King type operators. Filomat 27:173–181MathSciNetCrossRefzbMATHGoogle Scholar
  17. Rosenblum M (1994) Generalized Hermite polynomials and the Bose-like oscillator calculus. Oper Theory Adv Appl 73:369–396MathSciNetzbMATHGoogle Scholar
  18. Shisha O, Bond B (1968) The degree of convergence of linear positive operators. Proc Nat Acad Sci USA 60:1196–1200MathSciNetCrossRefzbMATHGoogle Scholar
  19. Sucu S (2014) Dunkl analogue of Szász operators. Appl Math Comput 244:42–48MathSciNetzbMATHGoogle Scholar
  20. Szász O (1950) Generalization of S. Bernstein’s polynomials to the infinite interval. J Res Nat Bur Stand 45:239–245MathSciNetCrossRefGoogle Scholar
  21. Wafi A, Rao N (2017a) Kantorovich form of generalized Szasz-type operastors with certain parameters using Charlier polynomials. Korean J Math 25(1):99–116MathSciNetGoogle Scholar
  22. Wafi A, Rao N (2017b) Stancu-Varient of Generalized-Baskakov operators. Filomat 31(9):2625–2632MathSciNetCrossRefGoogle Scholar
  23. Wafi A, Rao N (2017c) Szasz-Durrmeyer operators based on Dunkl analoguer. Complex Anal Oper Theory. Google Scholar
  24. Yüksel I, Ispir N (2006) Weighted approximation by a certain family of summation integral-type operators. Comput Math Appl 52(10–11):1463–1470MathSciNetCrossRefzbMATHGoogle Scholar

Copyright information

© Shiraz University 2017

Authors and Affiliations

  1. 1.Department of Mathematics, Faculty of Natural SciencesJamia Millia IslamiaNew DelhiIndia

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