Jump to ContentJump to Main Navigation
Show Summary Details
More options …

Georgian Mathematical Journal

Editor-in-Chief: Kiguradze, Ivan / Buchukuri, T.

Editorial Board: Kvinikadze, M. / Bantsuri, R. / Baues, Hans-Joachim / Besov, O.V. / Bojarski, B. / Duduchava, R. / Engelbert, Hans-Jürgen / Gamkrelidze, R. / Gubeladze, J. / Hirzebruch, F. / Inassaridze, Hvedri / Jibladze, M. / Kadeishvili, T. / Kegel, Otto H. / Kharazishvili, Alexander / Kharibegashvili, S. / Khmaladze, E. / Kiguradze, Tariel / Kokilashvili, V. / Krushkal, S. I. / Kurzweil, J. / Kwapien, S. / Lerche, Hans Rudolf / Mawhin, Jean / Ricci, P.E. / Tarieladze, V. / Triebel, Hans / Vakhania, N. / Zanolin, Fabio

IMPACT FACTOR 2018: 0.551

CiteScore 2018: 0.52

SCImago Journal Rank (SJR) 2018: 0.320
Source Normalized Impact per Paper (SNIP) 2018: 0.711

Mathematical Citation Quotient (MCQ) 2018: 0.27

See all formats and pricing
More options …
Ahead of print


Approximation by modified Jain–Baskakov operators

Vishnu Narayan MishraORCID iD: https://orcid.org/0000-0002-2159-7710
  • Corresponding author
  • Department of Mathematics, Indira Gandhi National Tribal University, Lalpur, Amarkantak, Anuppur, Madhya Pradesh 484-887; and L. 1627 Awadh Puri Colony Beniganj, Phase III, Opposite – Industrial Training Institute (I.T.I.), Ayodhya Main Road Faizabad 224-001, (Uttar Pradesh), India
  • orcid.org/0000-0002-2159-7710
  • Email
  • Other articles by this author:
  • De Gruyter OnlineGoogle Scholar
/ Preeti Sharma / Marius Mihai Birou
Published Online: 2019-02-16 | DOI: https://doi.org/10.1515/gmj-2019-2008


In the present paper, we discuss the approximation properties of Jain–Baskakov operators with parameter c. The paper deals with the modified forms of the Baskakov basis functions. Some direct results are established, which include the asymptotic formula, error estimation in terms of the modulus of continuity and weighted approximation. Also, we construct the King modification of these operators, which preserves the test functions e0 and e1. It is shown that these King type operators provide a better approximation order than some Baskakov–Durrmeyer operators for continuous functions defined on some closed intervals.

Keywords: Jain operators; Baskakov operators; King type operators; weighted approximation

MSC 2010: 41A25; 41A36


  • [1]

    U. Abel and O. Agratini, Asymptotic behaviour of Jain operators, Numer. Algorithms 71 (2016), no. 3, 553–565. Web of ScienceCrossrefGoogle Scholar

  • [2]

    D. Cárdenas-Morales, P. Garrancho and I. Raşa, Bernstein-type operators which preserve polynomials, Comput. Math. Appl. 62 (2011), no. 1, 158–163. CrossrefWeb of ScienceGoogle Scholar

  • [3]

    D. Cárdenas-Morales, P. Garrancho and I. Raşa, Asymptotic formulae via a Korovkin-type result, Abstr. Appl. Anal. 2012 (2012), Article ID 217464. Google Scholar

  • [4]

    N. Deo and N. Bhardwaj, Some approximation results for Durrmeyer operators, Appl. Math. Comput. 217 (2011), no. 12, 5531–5536. Web of ScienceGoogle Scholar

  • [5]

    R. A. DeVore and G. G. Lorentz, Constructive Approximation, Grundlehren Math. Wiss. 303, Springer, Berlin, 1993. Google Scholar

  • [6]

    D. K. Dubey and V. K. Jain, Rate of approximation for integrated Szasz–Mirakyan operators, Demonstr. Math. 41 (2008), no. 4, 879–886. Google Scholar

  • [7]

    A. D. Gadjiev, R. O. Efendiyev and E. İbikli, On Korovkin type theorem in the space of locally integrable functions, Czechoslovak Math. J. 53(128) (2003), no. 1, 45–53. Google Scholar

  • [8]

    A. D. Gadžiev, Theorems of the type of P. P. Korovkin’s theorems, Mat. Zametki 20 (1976), no. 5, 781–786. Google Scholar

  • [9]

    A. R. Gairola, Deepmala and L. N. Mishra, Rate of approximation by finite iterates of q-Durrmeyer operators, Proc. Nat. Acad. Sci. India Sect. A 86 (2016), no. 2, 229–234. CrossrefGoogle Scholar

  • [10]

    A. R. Gairola, Deepmala and L. N. Mishra, On the q-derivatives of a certain linear positive operators, Iran. J. Sci. Technol. Trans. A Sci. 42 (2018), no. 3, 1409–1417. CrossrefGoogle Scholar

  • [11]

    V. Gupta and G. C. Greubel, Moment estimations of new Szász–Mirakyan–Durrmeyer operators, Appl. Math. Comput. 271 (2015), 540–547. Web of ScienceGoogle Scholar

  • [12]

    M. Heilmann, Direct and converse results for operators of Baskakov–Durrmeyer type, Approx. Theory Appl. 5 (1989), no. 1, 105–127. Google Scholar

  • [13]

    G. C. Jain, Approximation of functions by a new class of linear operators, J. Aust. Math. Soc. 13 (1972), 271–276. CrossrefGoogle Scholar

  • [14]

    J. P. King, Positive linear operators which preserve x2, Acta Math. Hungar. 99 (2003), no. 3, 203–208. Google Scholar

  • [15]

    P. P. Korovkin, On convergence of linear positive operators in the space of continuous functions, Doklady Akad. Nauk SSSR (N. S.) 90 (1953), 961–964. Google Scholar

  • [16]

    Y. C. Kwun, A.-M. Acu, A. Rafiq, V. A. Radu, F. Ali and S. M. Kang, Bernstein-Stancu type operators which preserve polynomials, J. Comput. Anal. Appl. 23 (2017), no. 4, 758–770. Google Scholar

  • [17]

    V. N. Mishra, K. Khatri, L. N. Mishra and Deepmala, Inverse result in simultaneous approximation by Baskakov–Durrmeyer–Stancu operators, J. Inequal. Appl. 2013 (2013), Paper No. 586. Google Scholar

  • [18]

    P. Patel and V. N. Mishra, Jain–Baskakov operators and its different generalization, Acta Math. Vietnam. 40 (2015), no. 4, 715–733. Web of ScienceCrossrefGoogle Scholar

  • [19]

    N. Rao and A. Wafi, Stancu-variant of generalized Baskakov operators, Filomat 31 (2017), no. 9, 2625–2632. Web of ScienceCrossrefGoogle Scholar

  • [20]

    O. Shisha and B. Mond, The degree of convergence of sequences of linear positive operators, Proc. Natl. Acad. Sci. USA 60 (1968), 1196–1200. CrossrefGoogle Scholar

  • [21]

    O. Szasz, Generalization of S. Bernstein’s polynomials to the infinite interval, J. Res. Nat. Bur. Standards 45 (1950), 239–245. CrossrefGoogle Scholar

  • [22]

    S. Tarabie, On Jain-beta linear operators, Appl. Math. Inf. Sci. 6 (2012), no. 2, 213–216. Google Scholar

  • [23]

    S. Umar and Q. Razi, Approxiamtion of function by generalized Szász operators, Commun. Fac. Sci. L’Univ D’Ankara 34 (1985), 45–52. Google Scholar

  • [24]

    Z. Ziegler, Linear approximation and generalized convexity, J. Approx. Theory 1 (1968), 420–443. CrossrefGoogle Scholar

About the article

Received: 2017-04-15

Revised: 2017-10-26

Accepted: 2017-11-08

Published Online: 2019-02-16

Citation Information: Georgian Mathematical Journal, ISSN (Online) 1572-9176, ISSN (Print) 1072-947X, DOI: https://doi.org/10.1515/gmj-2019-2008.

Export Citation

© 2019 Walter de Gruyter GmbH, Berlin/Boston.Get Permission

Comments (0)

Please log in or register to comment.
Log in