Approximation properties and error estimation of q-Bernstein shifted operators

  • Mohammad MursaleenEmail author
  • Khursheed J. Ansari
  • Asif Khan
Original Paper


In the present paper, q-analogue of Lupaş Bernstein operators with shifted knots are introduced. First, some basic results for convergence of the introduced operators are established and then the rate of convergence by these operators in terms of the modulus of continuity are obtained. Further, a Voronovskaja type theorem and local approximation results for the said operators are studied. Error estimation tables are presented with respect to different parameters. We also show comparisons by some illustrative graphics for the convergence of operators to a function with the help of MATLAB R2018a.


Lupaş q-Bernstein shifted operators Rate of convergence Modulus of continuity Voronovskaja type theorem K-functional Local approximation 

Mathematics Subject Classification (2010)

41A10 41A25 41A36 



The authors extend their appreciation to the Deanship of Scientific Research at King Khalid University for funding this work through research groups program under Grant Number R.G.P.1/13/40.


  1. 1.
    Acar, T., Aral, A., Mohiuddine, S.A.: On Kantorovich modification of (p, q)-Baskakov operators. Journal of Inequalities and Applications 2016, 98 (2016)MathSciNetCrossRefzbMATHGoogle Scholar
  2. 2.
    Acar, T, Aral, A., Mohiuddine, S.: Approximation by bivariate (p,q)-Bernstein Kantorovich operators. Iran. J. Sci. Technol. Trans. Sci. 42, 655–662 (2018)MathSciNetCrossRefzbMATHGoogle Scholar
  3. 3.
    Acar, T.: (P,q)-generalization of Szasz-Mirakyan operators. Math. Methods Appl. Sci. 39, 2685–2695 (2016)MathSciNetCrossRefzbMATHGoogle Scholar
  4. 4.
    Altomare, F., Campiti, M.: Korovkin-Type Approximation Theory and Its Applications, de Gruyter Studies in Mathematics, vol. 17. Walter de Gruyter and Co., Berlin (1994)CrossRefzbMATHGoogle Scholar
  5. 5.
    Aral, A., Gupta, V., Agarwal, R.P.: Applications of Q-calculus in Operator Theory. Springer, New York (2013)CrossRefzbMATHGoogle Scholar
  6. 6.
    Bernstein, S.N.: Démonstation du théorème de Weierstrass fondée sur le calcul de probabilités. Commun. Soc. Math. Kharkow 13, 1–2 (1912–1913)Google Scholar
  7. 7.
    Cai, Q.-B., Zhou, G.: On (p,q)-analogue of Kantorovich type Bernstein-Stancu-Schurer operators. Appl. Math. Comput. 276, 12–20 (2016)MathSciNetzbMATHGoogle Scholar
  8. 8.
    Derriennic, M.M.: Modified Bernstein polynomials and Jacobi polynomials in q-calculus. Rend. Circ. Mat Palermo Serie II 76, 269–290 (2005)MathSciNetzbMATHGoogle Scholar
  9. 9.
    DeVore, R.A., Lorentz, G.G.: Constructive Approximation. Springer, Berlin (1993)CrossRefzbMATHGoogle Scholar
  10. 10.
    Dogru, O., Kanat, K.: On statistical approximation properties of the Kantorovich type Lupas operators. Math. Comput. Modell. 55, 1610–1620 (2012)MathSciNetCrossRefzbMATHGoogle Scholar
  11. 11.
    Gadjiev, A.D., Ghorbanalizadeh, A.M.: Approximation properties of a new type Bernstein-Stancu polynomials of one and two variables. Appl. Math. Comput. 216, 890–901 (2010)MathSciNetzbMATHGoogle Scholar
  12. 12.
    Gairola, A.R., Deepmala, Mishra, L.N.: Rate of approximation by finite iterates of q-Durrmeyer operators. Proc. Natl. Acad. Sci., India, Sect. A Phys. Sci. 86(2), 229–234 (2016)MathSciNetCrossRefzbMATHGoogle Scholar
  13. 13.
    Edely, O.H.H., Mohiuddine, S.A., Noman, A.K.: Korovkin type approximation theorems obtained through generalized statistical convergence. Appl. Math. Lett. 23, 1382–1387 (2010)MathSciNetCrossRefzbMATHGoogle Scholar
  14. 14.
    Kac, V., Cheung, P.: Quantum Calculus. Springer, New York (2002)CrossRefzbMATHGoogle Scholar
  15. 15.
    Kadak, U.: On weighted statistical convergence based on (p,q)-integers and related approximation theorems for functions of two variables. Jour. Math. Anal. Appl. 443, 752–764 (2016)MathSciNetCrossRefzbMATHGoogle Scholar
  16. 16.
    Kadak, U.: Weighted statistical convergence based on generalized difference operator involving (p,q)-Gamma function and its applications to approximation theorems. Jour. Math. Anal. Appl. 448, 1633–1650 (2017)MathSciNetCrossRefzbMATHGoogle Scholar
  17. 17.
    Kadak, U., Mishra, V.N., Pandey, S.: Chlodowsky type generalization of (p,q)-Szasz operators involving Brenke type polynomials. RACSAM 112(4), 1443–1462 (2018)MathSciNetCrossRefzbMATHGoogle Scholar
  18. 18.
    Khan, K., Lobiyal, D.K.: Bézier curves based on Lupas (p,q)-analogue of Bernstein functions in CAGD. Jour. Comput. Appl. Math. 317, 458–477 (2017)CrossRefzbMATHGoogle Scholar
  19. 19.
    Korovkin, P.P.: On convergence of linear operators in the space of continuous functions (Russian). Dokl. Akad. Nauk SSSR (N.S.) 90, 961–964 (1953)Google Scholar
  20. 20.
    Lorentz, G.G.: Bernstein Polynomials. University of Toronto Press, Toronto (1953)zbMATHGoogle Scholar
  21. 21.
    Lupaş, A.: A q-analogue of the Bernstein operator, Seminar on Numerical and Statistical Calculus. University of Cluj-Napoca 9, 85–92 (1987)zbMATHGoogle Scholar
  22. 22.
    Mishra, V.N., Khatri, K., Mishra, L.N.: Statistical approximation by Kantorovich type discrete q-Beta operators. Adv. Diff. Equ. 2013, 345 (2013)MathSciNetCrossRefzbMATHGoogle Scholar
  23. 23.
    Mishra, V.N., Patel, P., Mishra, L.N.: The integral type modification of jain operators and its approximation properties. Numer. Funct. Anal. Optim. 39(12), 1265–1277 (2018)MathSciNetCrossRefzbMATHGoogle Scholar
  24. 24.
    Mursaleen, M., Ansari, K.J., Khan, A.: On (p,q) -analogue of Bernstein operators. Appl. Math. Comput. 266, 874–882 (2015). [Erratum: Appl. Math. Comput., 278 (2016) 70-71]MathSciNetzbMATHGoogle Scholar
  25. 25.
    Mursaleen, M., Khan, A.: Statistical approximation properties of modified q-Stancu-Beta operators. Bull. Malaysian Math. Sci. Soc.(2) 36, 683–690 (2013)MathSciNetzbMATHGoogle Scholar
  26. 26.
    Mursaleen, M., Khan, F., Khan, A.: Statistical approximation for new positive linear operators of Lagrange type. Appl. Math Comput. 232, 548–558 (2014)MathSciNetzbMATHGoogle Scholar
  27. 27.
    Phillips, G.M.: Bernstein polynomials based on the q -integers. Ann. Numer. Math. 4, 511–518 (1997)MathSciNetzbMATHGoogle Scholar
  28. 28.
    Stancu, D.D.: Approximation of functions by a new class of linear polynomial operators. Rev. Roumaine Math. Pure Appl. 13, 1173–1194 (1968)MathSciNetzbMATHGoogle Scholar
  29. 29.
    Rao, N., Wafi, A.: (P,q)-bivariate-Bernstein-Chlodowsky operators. Filomat 32, 369–378 (2018)MathSciNetCrossRefGoogle Scholar

Copyright information

© Springer Science+Business Media, LLC, part of Springer Nature 2019

Authors and Affiliations

  • Mohammad Mursaleen
    • 1
    Email author
  • Khursheed J. Ansari
    • 2
  • Asif Khan
    • 1
  1. 1.Department of MathematicsAligarh Muslim UniversityAligarhIndia
  2. 2.Department of Mathematics, College of ScienceKing Khalid UniversityAbhaSaudi Arabia

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