Advertisement

Approximation by Jakimovski–Leviatan operators of Durrmeyer type involving multiple Appell polynomials

  • Khursheed J. Ansari
  • M. Mursaleen
  • Shagufta Rahman
Original Paper

Abstract

In the present paper, we introduce Jakimovski–Leviatan–Durrmeyer type operators involving multiple Appell polynomial. First, we investigate Korovkin type approximation theorem and rate of convergence by using usual modulus of continuity and class of Lipschitz function. Next, we study the convergence of these operators in weighted space of functions and estimate the approximation properties. We have also established Voronovskaja type asymptotic formula. Furthermore, we obtain statistical approximation properties of these operators with the help of universal Korovkin type statistical approximation theorem. Some graphical examples for the convergence of our operators towards some functions are given. At the end, we have computed error estimation as our numerical example.

Keywords

Multiple Apple polynomial Durrmeyer operators Jakimovski–Leviatan operators Modulus of continuity Statistical approximation 

Mathematics Subject Classification

41A10 41A28 41A36 

Notes

Acknowledgements

The first author would like to express his gratitude to King Khalid University, Abha, Saudi Arabia for providing administrative and technical support.

References

  1. 1.
    Altomare, F., Campiti, M.: Korovkin Type Approximation Theory and its Applications, De Gruyter Studies in Mathematics, vol. 17. Walter de Gruyter, Berlin (1994)CrossRefzbMATHGoogle Scholar
  2. 2.
    Duman, O., Orhan, C.: Statistical approximation by positive linear operators. Stud. Math. 161(2), 187–197 (2004)MathSciNetCrossRefzbMATHGoogle Scholar
  3. 3.
    Fast, H.: Sur la convergence statistique. Colloq. Math. 2, 241–244 (1951)MathSciNetCrossRefzbMATHGoogle Scholar
  4. 4.
    Gadjiev, A.D.: The convergence problem for a sequence of positive linear operators on bounded sets and theorems analogous to that of P.P. Korovkin, Dokl. Akad. Nauk SSSR 218 (5) (1974) (Transl. in Soviet Math. Dokl. 15(5), 1433–1436 (1974))Google Scholar
  5. 5.
    Gadjiev, A.D.: On P.P. Korovkin type theorems, Mat. Zametki 20:781–786 (1976) (Transl. Math. Notes 5–6, 995–998 (1978))Google Scholar
  6. 6.
    Gadjiev, A.D., Orhan, C.: Some approximation theorems via statistical convergence. Rocky Mt. J. Math. 32, 129–138 (2002)MathSciNetCrossRefzbMATHGoogle Scholar
  7. 7.
    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), 229234 (2016).  https://doi.org/10.1007/s40010-016-0267-z
  8. 8.
    Gairola, A.R., Deepmala, Mishra, L.N.: On the \(q\)-derivatives of a certain linear positive operators. Iran. J. Sci. Technol. Trans. A Sci. (2017).  https://doi.org/10.1007/s40995-017-0227-8
  9. 9.
    Gandhi, R.B., Deepmala, V.N.Mishra: Local and global results for modified Szász–Mirakjan operators. Math. Method Appl. Sci. 40(7), 2491–2504 (2017).  https://doi.org/10.1002/mma.4171 CrossRefzbMATHGoogle Scholar
  10. 10.
    Jakimovski, A., Leviatan, D.: Generalized Sźasz operators for the approximation in the infinite interval. Mathematica (Cluj) 11, 97–103 (1969)MathSciNetzbMATHGoogle Scholar
  11. 11.
    Karaisa, A.: Approximation by Durrmeyer type Jakimoski Leviatan operators. Math. Method Appl. Sci. 39(9), 2401–2410 (2016)MathSciNetCrossRefzbMATHGoogle Scholar
  12. 12.
    Khatri, K., Mishra, V.N.: Generalized Szász–Mirakyan operators involving Brenke type polynomials. Appl. Math. Comput. 324, 228–238 (2018)MathSciNetGoogle Scholar
  13. 13.
    Lee, D.W.: On multiple Appell polynomials. Proc. Am. Math. Soc. 139(6), 2133–2141 (2011)MathSciNetCrossRefzbMATHGoogle Scholar
  14. 14.
    Mishra, V.N., Gandhi, R.B.: Simultaneous approximation by Sźasz–Mirakjan–Stancu–Durrmeyer type operators. Period. Math. Hung. 74(1), 118–127 (2017).  https://doi.org/10.1007/s10998-016-0145-0 CrossRefzbMATHGoogle Scholar
  15. 15.
    Mishra, V.N., Khatri, K., Mishra, L.N.: Deepmala, Inverse result in simultaneous approximation by Baskakov–Durrmeyer–Stancu operators. J. Inequal. Appl. (2013).  https://doi.org/10.1186/1029-242X-2013-586
  16. 16.
    Mursaleen, M., Al-Abied, A., Ansari, K.J.: Rate of convergence of Chlodowsky type Durrmeyer Jakimovski–Leviatan operators. Tbil. Math. J. 10(2), 173–184 (2017)MathSciNetzbMATHGoogle Scholar
  17. 17.
    Mursaleen, M., Ansari, K.J.: Approximation by generalized Szász operators involving Sheffer polynomials (2015). arXiv preprint: arXiv:1601.00675
  18. 18.
    Mursaleen, M., Ansari, K.J.: Some approximation results on two parametric \(q\)-Stancu-Beta operators. Bull. Malays. Math. Sci. Soc. (2017).  https://doi.org/10.1007/s40840-017-0499-3 zbMATHGoogle Scholar
  19. 19.
    Mursaleen, M., Ansari, K.J.: On Chlodowsky variant of Szász operators by Brenke type polynomials. Appl. Math. Comput. 271, 991–1003 (2015)MathSciNetGoogle Scholar
  20. 20.
    Mursaleen, M., Ansari, K.J., Nasiruzzaman, M.: Approximation by \(q\)-analogue of Jakimovski–Leviatan operators involving \(q\)-Appell polynomials. Iran. J. Sci. Technol. Trans. A Sci. (2017).  https://doi.org/10.1007/s40995-017-0331-9
  21. 21.
    Šalát, T.: On statistically convergent sequences of real numbers. Math. Slovaca 30, 139–150 (1980)MathSciNetzbMATHGoogle Scholar
  22. 22.
    Sźasz, O.: Generalization of S. Bernsteins polynomials to the infinite interval. J. Res. Natl. Bur. Stand. 97, 239–245 (1950)MathSciNetCrossRefGoogle Scholar
  23. 23.
    Varma, S.: On a generalization of Sźasz operators by multiple Appell polynomials. Stud. Univ. Babeş-Bolyai Math. 58(3), 361–369 (2013)zbMATHGoogle Scholar
  24. 24.
    Ÿuksel, I., Ispir, N.: Weighted approximation by a certain family of summation integral-type operators. Comput. Math. Appl. 52(10–11), 1463–1470 (2006)Google Scholar

Copyright information

© Springer-Verlag Italia S.r.l., part of Springer Nature 2018

Authors and Affiliations

  • Khursheed J. Ansari
    • 1
  • M. Mursaleen
    • 2
  • Shagufta Rahman
    • 2
  1. 1.Department of Mathematics, College of ScienceKing Khalid UniversityAbhaSaudi Arabia
  2. 2.Department of MathematicsAligarh Muslim UniversityAligarhIndia

Personalised recommendations