Further results concerning some general Durrmeyer type operators

  • Tarul Garg
  • Ana Maria AcuEmail author
  • Purshottam Narain Agrawal
Original Paper


The present paper deals with the approximation properties of the Miheşan-Durrmeyer type operators due to Kajla (Adv Oper Theory 2:162–178, 2017) which include well-known operators like Phillips operators (Ann Math 59:325–356, 1954). and Pǎltǎnea operators (Carpathian J Math 24:378–385, 2008) as particular cases. We establish the degree of approximation of the operators in a weighted space and then extend the study to the approximation in a weighted space and the order of convergence of the operators by means of the Ditzian–Totik modulus of smoothness, quantitative and Grüss Voronovskaya type approximation. We construct a variant of our operator that preserves the test functions \(e_0\) and \(e_2\) and show that the new operators present a better rate of convergence than the original ones. Next we define an operator which preserves the functions \(e_0\) and \(e^{-x}\) and show by numerical examples and illustrations that at least one of the two newly constructed operators presents a better rate of convergence than the original operator.


Ditzian–Totik modulus of smoothness Rate of convergence Weighted space Voronovskaya type theorem 

Mathematics Subject Classification

41A10 41A25 41A36 41A60 



The authors are extremely grateful to the learned reviewers for a very critical reading of the paper and making valuable comments and suggestions leading to a better presentation of the paper. The first author is thankful to “The Ministry of Human Resource and Development”, India for the financial support to carry out her research work and the work of the second author was financed from Lucian Blaga University of Sibiu research grant LBUS-IRG-2018-04.


  1. 1.
    Acar, T.: Quantitative \(q\)-Voronovskaya and \(q\)-Grüss-Voronovskaya-type results for \(q\)-Szász operators. Georgian Math. J. 23(4), 459–468 (2016)MathSciNetzbMATHGoogle Scholar
  2. 2.
    Acar, T., Aral, A., Gonska, H.: On Szász-Mirakyan operators preserving \(e^{2ax}\), \(a>0\). Mediterr. J. Math. 6, 14 (2017)zbMATHGoogle Scholar
  3. 3.
    Acar, T., Aral, A., Rasa, I.: The new forms of Voronovskaya’s theorem in weighted spaces. Positivity 20(1), 25–40 (2016)MathSciNetzbMATHGoogle Scholar
  4. 4.
    Acu, A.M., Gonska, H., Rasa, I.: Grüss-type and Ostrowski-type inequalities in approximation theory. Ukrainian Math. J. 63(6), 843–864 (2011)MathSciNetzbMATHGoogle Scholar
  5. 5.
    Agratini, O.: Linear operators that preserve some test functions. Int. J. Math. Math. Sci. 94136, 11 (2006)MathSciNetzbMATHGoogle Scholar
  6. 6.
    Ditzian, Z., Totik, V.: Moduli of smoothness. Springer series in computational mathematics, 9. Springer, New York (1987)zbMATHGoogle Scholar
  7. 7.
    Erençin, A., Rasa, I.: Voronovskaya type theorems in weighted spaces. Numer. Funct. Anal. Optim. 37(12), 1517–1528 (2016)MathSciNetzbMATHGoogle Scholar
  8. 8.
    Gadžiev, A.D.: Positive linear operators in weighted spaces of functions of several variables. (Russian) Izv. Akad. Nauk Azerbaidzhan. SSR Ser. Fiz.-Tekhn. Mat. Nauk 1(4), 32–37 (1980)MathSciNetGoogle Scholar
  9. 9.
    Gadžiev, A.D., Hacısalihoǧlu, H.: Convergence of the sequences of linear positive operators. Ankara University Press, Ankara (1995)Google Scholar
  10. 10.
    Gairola, A.R., Deepmala, Mishra, L.N.: Rate of approximation by finite iterates of \(q\)-Durrmeyer operators. Proc. Nat. Acad. Sci. India Sect. A 86(2), 229–234 (2016)MathSciNetzbMATHGoogle Scholar
  11. 11.
    Gairola, A.R., Deepmala, Mishra, L.N.: On the \(q\)-derivatives of a certain linear positive operators. Iran. J. Sci. Technol. Trans. A Sci 42(3), 1409–1417 (2018)MathSciNetzbMATHGoogle Scholar
  12. 12.
    Gal, S.G., Gonska, H.: Grüss and Grüss-Voronovskaya-type estimates for some Bernstein-type polynomials of real and complex variables. Jaen J. Approx. 7(1), 97–122 (2015)MathSciNetzbMATHGoogle Scholar
  13. 13.
    Gandhi, R.B., Deepmala, Mishra, V.N.: Local and global results for modified Szsz-Mirakjan operators. Math. Methods Appl. Sci. 40(7), 2491–2504 (2017)MathSciNetzbMATHGoogle Scholar
  14. 14.
    Gonska, H., Tachev, G.: Grüss-type inequalities for positive linear operators with second order moduli. Mat. Vesnik 63(4), 247–252 (2011)MathSciNetzbMATHGoogle Scholar
  15. 15.
    Grüss, G.: Über das Maximum des absoluten Betrages von \(\frac{1}{{b - a}}\int \limits _a^b f(x)g(x)dx - \frac{1}{(b-a)^2}\int \limits _a^b f(x) dx \int \limits _a^b g(x) dx\). (German) Math. Z 39(1), 215–226 (1935)MathSciNetzbMATHGoogle Scholar
  16. 16.
    Gupta, V., Aral, A.: A note on Szász-Mirakyan-Kantorovich type operators preserving \(e^{-x}\). Positivity 2, 5 (2017)zbMATHGoogle Scholar
  17. 17.
    Gupta, V., Tachev, G.: On approximation properties of Phillips operators preserving exponential functions. Mediterr. J. Math. 14(4), 14:177 (2017)MathSciNetzbMATHGoogle Scholar
  18. 18.
    İspir, N.: On modified Baskakov operators on weighted spaces. Turkish J. Math. 25(3), 355–365 (2001)MathSciNetzbMATHGoogle Scholar
  19. 19.
    İspir, N., Atakut, Ç.: Approximation by modified Szász–Mirakjan operators on weighted spaces. Proc. Indian Acad. Sci. Math. Sci. 112(4), 571–578 (2002)MathSciNetzbMATHGoogle Scholar
  20. 20.
    Jain, G.C., Pethe, S.: On the generalizations of Bernstein and Száz–Mirakyan operators. Nanta Math. 10(2), 185–193 (1977)MathSciNetzbMATHGoogle Scholar
  21. 21.
    Kajla, A.: Direct estimates of certain Miheşan–Durrmeyer type operators. Adv. Oper. Theory 2(2), 162–178 (2017)MathSciNetzbMATHGoogle Scholar
  22. 22.
    Kajla, A., Agrawal, P.N.: Approximation properties of Szász type operators based on Charlier polynomials. Turkish J. Math. 39(6), 990–1003 (2015)MathSciNetzbMATHGoogle Scholar
  23. 23.
    Lenze, B.: On Lipschitz-type maximal functions and their smoothness spaces. Nederl. Akad. Wetensch. Indag. Math. 50(1), 53–63 (1988)MathSciNetzbMATHGoogle Scholar
  24. 24.
    Miheşan, V.: Gamma approximating operators. Creat. Math. Inform. 17(3), 466–472 (2008)MathSciNetzbMATHGoogle Scholar
  25. 25.
    Mishra, V.N., Khatri, K., Mishra, L.N.: On simultaneous approximation for Baskakov–Durrmeyer–Stancu type operators. Ultra Sci. 24(3A), 567–577 (2012)zbMATHGoogle Scholar
  26. 26.
    Mishra, V.N., Khatri, K., Mishra, L.N.: Statistical approximation by Kantorovich-type discrete \(q\)-beta operators. Adv. Differ. Equ. 345, 15 (2013)MathSciNetzbMATHGoogle Scholar
  27. 27.
    Mishra, V.N., Khatri, K., Mishra, L.N.: Deepmala: Inverse result in simultaneous approximation by Baskakov–Durrmeyer–Stancu operators. J. Inequal. Appl. 586, 11 (2013)zbMATHGoogle Scholar
  28. 28.
    Mishra, V.N., Patel, P., Mishra, L.N.: The Integral type Modification of Jain Operators and its Approximation Properties. Numerical Functional Analysis and Optimization (2018).
  29. 29.
    Mond, B.: On the degree of approximation by linear positive operators. J. Approx. Theory 18(3), 304–306 (1976)MathSciNetzbMATHGoogle Scholar
  30. 30.
    Neer, T., Agrawal, P.N., Araci, S.: Stancu–Durrmeyer type operators based on \(q\)-integers. Appl. Math. Inf. Sci. 11(3), 1–9 (2017)MathSciNetGoogle Scholar
  31. 31.
    Pǎltǎnea, R.: Modified Szász–Mirakjan operators of integral form. Carpathian J. Math. 24(3), 378–385 (2008)zbMATHGoogle Scholar
  32. 32.
    Phillips, R.S.: An inversion formula for Laplace transforms and semi-groups of linear operators. Ann. Math. 59(2), 325–356 (1954)MathSciNetzbMATHGoogle Scholar
  33. 33.
    Singh, K.K., Gairola, A.R.: Approximation theorems for \(q-\) analouge of a linear positive operator by A. Lupas. Int. J. Anal. Appl. 12(1), 30–37 (2016)zbMATHGoogle Scholar
  34. 34.
    Yüksel, I., İspir, N.: Weighted approximation by a certain family of summation integral-type operators. Comput. Math. Appl. 52(10–11), 1463–1470 (2007)MathSciNetzbMATHGoogle Scholar

Copyright information

© The Royal Academy of Sciences, Madrid 2019

Authors and Affiliations

  1. 1.Department of MathematicsIndian Institute of Technology RoorkeeRoorkeeIndia
  2. 2.Department of Mathematics and InformaticsLucian Blaga University of SibiuSibiuRomania

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