Advertisement

Complex Analysis and Operator Theory

, Volume 13, Issue 3, pp 1537–1556 | Cite as

Approximation by a Class of q-Beta Operators of the Second Kind Via the Dunkl-Type Generalization on Weighted Spaces

  • H. M. SrivastavaEmail author
  • M. Mursaleen
  • Md. Nasiruzzaman
Article
  • 42 Downloads

Abstract

The aim of the present article is to study the approximation and other related properties of a class of q-Szász–Beta type operators of the second kind. In this context, we construct the class of q-Szász–Beta type operators of the second kind, which are generated by means of the exponential functions of the basic (or q-) calculus via the Dunkl-type generalization. In order to get a uniform convergence on weighted spaces, we obtain Korovkin-type approximation theorems involving local approximations and weighted approximations, the rate of convergence in terms of the classical, the second-order and the weighted moduli of continuity, as well as a set of direct theorems. Relevant connection of the results presented in this article with those in earlier works is also indicated.

Keywords

Basic (or q-) calculus Basic (or q-) integers Basic (or q-) Beta functions Basic (or q-) exponential functions Dunkl’s analogue Generalized exponential functions Szász operator Modulus of continuity Peetre’s K-functional Weighted modulus of continuity Korovkin-type approximation theorems 

Mathematics Subject Classification

Primary 41A25 41A36 Secondary 33C45 

Notes

References

  1. 1.
    Atakut, Ç., İspir, N.: Approximation by modified Szász–Mirakjan operators on weighted spaces. Proc. Indian Acad. Sci. Math. Sci. 112, 571–578 (2002)MathSciNetCrossRefzbMATHGoogle Scholar
  2. 2.
    Aral, A., Gupta, V., Agarwal, R.P.: Applications of \(q\)-Calculus in Operator Theory. Springer, New York (2013)CrossRefzbMATHGoogle Scholar
  3. 3.
    Ben Cheikh, Y., Gaied, Y., Zaghouani, M.: A \(q\)-Dunkl-classical \(q\)-Hermite type polynomials. Georgian Math. J. 21, 125–137 (2014)MathSciNetCrossRefzbMATHGoogle Scholar
  4. 4.
    De Sole, A., Kac, V.G.: On integral representations of \(q\)-gamma and \(q\)-beta functions. Atti Accad. Naz. Lincei Cl. Sci. Fis. Mat. Natur. Rend. Lincei (Ser. 9) Mat. Appl. 16, 11–29 (2005)MathSciNetzbMATHGoogle Scholar
  5. 5.
    De Vore, R.A., Lorentz, G.G.: Constructive Approximation. Springer, Berlin (1993)Google Scholar
  6. 6.
    Gupta, V., Agrawal, P.N.: Rate of convergence of Szász–Beta operators based on \(q\)-integers. Demonstr. Math. 50, 130–143 (2017)MathSciNetCrossRefzbMATHGoogle Scholar
  7. 7.
    Gupta, V., Noor, M.A.: Convergence of derivatives for certain mixed Szász–Beta operators. J. Math. Anal. Appl. 321, 1–9 (2006)MathSciNetCrossRefzbMATHGoogle Scholar
  8. 8.
    İçöz, G., Çekim, B.: Dunkl generalization of Szász operators via \(q\)-calculus. J. Inequal. Appl. 2015(Article ID 284), 1–11 (2015)zbMATHGoogle Scholar
  9. 9.
    İçöz, G., Çekim, B.: Stancu type generalization of Dunkl analogue of Szász-Kamtrovich operators. Math. Methods Appl. Sci. 39, 1803–1810 (2016)MathSciNetCrossRefzbMATHGoogle Scholar
  10. 10.
    Jackson, F.H.: On \(q\)-definite integrals. Quart. J. Pure Appl. Math. 41, 193–203 (1910)zbMATHGoogle Scholar
  11. 11.
    Kac, V., Cheung, P.: Quantum Calculus. Universitext. Springer, Berlin (2002)CrossRefzbMATHGoogle Scholar
  12. 12.
    Korovkin, P.P.: Convergence of linear positive operators in the spaces of continuous functions (in Russian). Dokl. Akad. Nauk. SSSR (N. S.) 90, 961–964 (1953)Google Scholar
  13. 13.
    Milovanović, G.V., Mursaleen, M., Nasiruzzaman, Md: Modified Stancu type Dunkl generalization of Szász–Kantorovich operators. Rev. Real Acad. Cienc. Exactas Fís. Natur. Ser. A Mat. (RACSAM) 112, 135–151 (2018)CrossRefzbMATHGoogle Scholar
  14. 14.
    Mursaleen, M., Nasiruzzaman, Md, Alotaibi, A.M.: On modified Dunkl generalization of Szász operators via \(q\)-calculus. J. Inequal. Appl. 2017(Article ID 38), 1–12 (2017)zbMATHGoogle Scholar
  15. 15.
    Mursaleen, M., Nasiruzzaman, Md: Dunkl generalization of Kantorovich type Szász–Mirakjan operators via \(q\)-calculus. Asian Eur. J. Math. 10(4, Article ID 1750077), 1–17 (2017)CrossRefzbMATHGoogle Scholar
  16. 16.
    Peetre, J.: A Theory of Interpolation of Normed Spaces. Noteas de Mathematica, vol. 39. Instituto de Mathemática Pura e Applicada, Conselho Nacional de Pesquidas, Rio de Janeiro (1968)zbMATHGoogle Scholar
  17. 17.
    Rao, N., Wafi, A., Acu, A.M.: \(q\)-Szász–Durrmeyer type operators based on Dunkl analogue. Complex Anal. Oper. Theory (2018).  https://doi.org/10.1007/s11785-018-0816-3
  18. 18.
    Rosenblum, M.: Generalized Hermite polynomials and the Bose-like oscillator calculus. Oper. Theory Adv. Appl. 73, 369–396 (1994)MathSciNetzbMATHGoogle Scholar
  19. 19.
    Srivastava, H.M., Jena, B.B., Paikray, S.K., Misra, U.K.: A certain class of weighted statistical convergence and associated Korovkin-type approximation theorems involving trigonometric functions. Math. Meth. Appl. Sci. 41, 671–683 (2018)MathSciNetzbMATHGoogle Scholar
  20. 20.
    Srivastava, H.M., Mursaleen, M., Alotaibi, A.M., Nasiruzzaman, Md, Al-Abied, A.: Some approximation results involving the \(q\)-Szász–Mirakjan–Kantorovich type operators via Dunkl’s generalization. Math. Meth. Appl. Sci. 40, 5437–5452 (2017)CrossRefzbMATHGoogle Scholar
  21. 21.
    Sucu, S.: Dunkl analogue of Szász operators. Appl. Math. Comput. 244, 42–48 (2014)MathSciNetzbMATHGoogle Scholar
  22. 22.
    Szász, O.: Generalization of S. Bernstein’s polynomials to the infinite interval. J. Res. Nat. Bur. Stand. 45, 239–245 (1950)MathSciNetCrossRefGoogle Scholar

Copyright information

© Springer Nature Switzerland AG 2019

Authors and Affiliations

  1. 1.Department of Mathematics and StatisticsUniversity of VictoriaVictoriaCanada
  2. 2.Department of Medical Research, China Medical University HospitalChina Medical UniversityTaichungPeople’s Republic of China
  3. 3.Department of MathematicsAligarh Muslim UniversityAligarhIndia
  4. 4.Department of Computer Science (SEST)Jamia Hamdard UniversityHamdard NagarIndia

Personalised recommendations