Approximation properties of Lupas–Kantorovich operators based on Polya distribution

  • P. N. Agrawal
  • Nurhayat Ispir
  • Arun Kajla


In this paper, we introduce Kantorovich modification of the operators considered by Lupas and Lupas (Stud Univ Babes-Bolyai Math 32(4):61–69, 1987) based on Polya distribution and study Voronovskaja type asymptotic formula, local approximation, pointwise estimates and global approximation results. In the last section, we consider the bivariate generalization of these operators and discuss the rate of convergence. We also illustrate the convergence of these operators to some functions by graphics in Maple for both one and two dimensional cases and also estimate the error in the approximation by giving numerical examples for the bivariate case.


Asymptotic formula Local approximation Global approximation Polya distribution 

Mathematics Subject Classification

41A25 26A15 



The authors are extremely grateful to the reviewers for a very careful reading of the manuscript and making valuable suggestions leading to a better presentation of the paper. The last author is thankful to the “University Grants Commission” India for financial support to carry out the above research work.


  1. 1.
    Abel, U.: Asymptotic approximation with Kantorovich polynomial. Approx. Theory Appl. 14(3), 106–116 (1998)MathSciNetzbMATHGoogle Scholar
  2. 2.
    Agrawal, P.N., Ispir, N., Kajla, A.: Approximation properites of Bezier-summation-integral type operators based on Polya–Bernstein functins. Appl. Math. Comput. 259, 533–539 (2015)MathSciNetGoogle Scholar
  3. 3.
    Altomare, F., Cappelletti Montano, M., Leonessa, V.: On a generalization of Szász–Mirakjan–Kantorovich operators. Results Math. 63, 837–863 (2013)MathSciNetCrossRefzbMATHGoogle Scholar
  4. 4.
    Anastassiou, G.A., Gal, S.: Approximation Theory: Moduli of Continuity and Global Smoothness Preservation. Birkhäuser, Boston (2000)CrossRefzbMATHGoogle Scholar
  5. 5.
    Bǎrbosu, D., Muraru, C.V.: Approximating \(B\)-continuous functions using GBS operators of Bernstein–Schurer–Stancu type based on \(q\)-integers. Appl. Math. Comput. 259, 80–87 (2015)MathSciNetGoogle Scholar
  6. 6.
    Butzer, P.L.: Summability of generalized Bernstein polynomials. Duke Math. J. 22, 617–623 (1955)MathSciNetCrossRefzbMATHGoogle Scholar
  7. 7.
    Butzer, P.L.: On the extensions of Bernstein polynomials to the infinite interval. Proc. Am. Math. Soc. 5, 547–553 (1954)MathSciNetCrossRefzbMATHGoogle Scholar
  8. 8.
    Butzer, P. L., Berens, H.: Semi-Groups of Operators and Approximation, pp. xi+318. Springer, New York (1967)Google Scholar
  9. 9.
    Cal, J., Valle, A.M.: A generalization of Bernstein–Kantorovich operators. J. Math. Anal. Appl. 252, 750–766 (2000)MathSciNetCrossRefzbMATHGoogle Scholar
  10. 10.
    Cárdenas-Morales, D., Gupta, V.: Two families of Bernstein–Durrmeyer type operators. Appl. Math. Comput. 248, 342–353 (2014)MathSciNetzbMATHGoogle Scholar
  11. 11.
    Devore, R.A., Lorentz, G.G.: Constructive Approximation, Grundlehren der Mathematischen Wissenschaften, vol. 303. Springer-, Berlin, Heidelberg, New York, London (1993)Google Scholar
  12. 12.
    Ditzian, Z., Totik, V.: Moduli of Smoothness. Springer, New York (1987)CrossRefzbMATHGoogle Scholar
  13. 13.
    Doǧru, O., Gupta, V.: Korovkin-type approximation propeties of bivariate q-Meyer–König and Zeller operators. Calcolo 43, 51–63 (2006)MathSciNetCrossRefGoogle Scholar
  14. 14.
    Duman, O., Özarslan, M.A., Della Vecchia, B.: Modified Szász–Mirakjan–Kantorovich operators preserving linear functions. Turk. J. Math. 33, 151–158 (2009)zbMATHGoogle Scholar
  15. 15.
    Erençin, A., Büyükdurakoĝlu, S.: A modiffcation of generalized Baskakov-Kantorovich operators. Stud. Univ. Babes-Bolyai, Math. 59, 351–364 (2014)MathSciNetGoogle Scholar
  16. 16.
    Gonska, H., Heilmann, M., Rasa, I.: Kantorovich operators of order \(k\). Numer. Funct. Anal. Optim. 32, 717–738 (2011)MathSciNetCrossRefzbMATHGoogle Scholar
  17. 17.
    Gupta, V., Rassias, T.M.: Lupas–Durrmeyer operators based on Polya distribution. Banach J. Math. Anal. 8(2), 146–155 (2014)MathSciNetCrossRefzbMATHGoogle Scholar
  18. 18.
    Gurdek, M., Rempulska, L., Skorupka, M.: The Baskakov operators for the functions of two variables. Collect. Math. 50(3), 298–302 (1999)MathSciNetzbMATHGoogle Scholar
  19. 19.
    Kantorovich, L.V.: Sur certains d veloppements suivant les polynSmials de la formede S. Bernstein I, II. C. R. Acad. Sci. USSR, 20, A, 563–568, 595–600 (1930)Google Scholar
  20. 20.
    Lenze, B.: On Lipschitz-type maximal functions and their smoothness spaces. Nederl. Akad. Wetensch. Indag. Math. 50(1), 53–63 (1988)MathSciNetCrossRefzbMATHGoogle Scholar
  21. 21.
    Liu, X.L., Xi, K.Y., Guo, S.S.: Weighted approximation by Szász–Kantorovich operators. J. Central China Normal Univ. Natur. Sci. 36(3), 269–272 (2002)MathSciNetzbMATHGoogle Scholar
  22. 22.
    Lupas, L., Lupas, A.: Polynomials of binomial type and approximation operators. Stud. Univ. Babes-Bolyai Math. 32(4), 61–69 (1987)MathSciNetzbMATHGoogle Scholar
  23. 23.
    Miclaus, D.: The revision of some results for Bernstein Stancu type operators. Carpath. J. Math. 28(2), 289–300 (2012)MathSciNetzbMATHGoogle Scholar
  24. 24.
    Mihesan, V.: Uniform approximation with positive linear operators generated by generalised Baskakov method. Autom. Comput. Appl. Math. 7, 34–37 (1998)MathSciNetGoogle Scholar
  25. 25.
    Mishra, V. N., Khatri, K., Mishra, L. N.: Statistical approximation by Kantorovich-type discrete q- Beta operators. Adv. Differ. Equ. 2013, 1–15 (2013). (article 345)Google Scholar
  26. 26.
    Özarslan, M.A., Duman, O.: Local approximation behaviour of modified SMK operators. Miskolc Math. Notes 11(1), 87–89 (2010)MathSciNetzbMATHGoogle Scholar
  27. 27.
    Paltanea, R.: A note on generalized Bernstein–Kantorovich operators. Bull. Transilv. Univ. Braşov. Ser. 6(55), 27–32 (2013)MathSciNetzbMATHGoogle Scholar
  28. 28.
    Totik, V.: Uniform approximation by Szász–Mirakjan type operators. Acta Math. Hungar. 41, 291–307 (1983)MathSciNetCrossRefzbMATHGoogle Scholar
  29. 29.
    Totik, V.: Approximation by Szász–Mirakjan–Kantorovich operators in \(L^p (p > 1)\). Anal. Math. 9, 147–167 (1983)MathSciNetCrossRefzbMATHGoogle Scholar
  30. 30.
    Wafi, A., Khatoon, S.: Convergence and Voronovskaja-type theorems for derivatives of generalized Baskakov operators. Cent. Eur. J. Math. 6, 325–334 (2008)MathSciNetCrossRefzbMATHGoogle Scholar

Copyright information

© Springer-Verlag Italia 2015

Authors and Affiliations

  1. 1.Department of MathematicsIndian Institute of Technology RoorkeeRoorkeeIndia
  2. 2.Department of Mathematics, Faculty of SciencesGazi UniversityAnkaraTurkey

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