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Approximation by modified Jain–Baskakov operators

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Published/Copyright: February 16, 2019
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Georgian Mathematical Journal
From the journal Georgian Mathematical Journal Volume 27 Issue 3

Abstract

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

References

[1] U. Abel and O. Agratini, Asymptotic behaviour of Jain operators, Numer. Algorithms 71 (2016), no. 3, 553–565. 10.1007/s11075-015-0009-3Search in Google 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. 10.1016/j.camwa.2011.04.063Search in Google 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. 10.1155/2012/217464Search in Google Scholar

[4] N. Deo and N. Bhardwaj, Some approximation results for Durrmeyer operators, Appl. Math. Comput. 217 (2011), no. 12, 5531–5536. 10.1016/j.amc.2010.12.026Search in Google Scholar

[5] R. A. DeVore and G. G. Lorentz, Constructive Approximation, Grundlehren Math. Wiss. 303, Springer, Berlin, 1993. 10.1007/978-3-662-02888-9Search in 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. 10.1515/dema-2008-0415Search in 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. 10.1023/A:1022967223553Search in Google Scholar

[8] A. D. Gadžiev, Theorems of the type of P. P. Korovkin’s theorems, Mat. Zametki 20 (1976), no. 5, 781–786. 10.1007/BF01146928Search in 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. 10.1007/s40010-016-0267-zSearch in Google 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. 10.1007/s40995-017-0227-8Search in Google Scholar

[11] V. Gupta and G. C. Greubel, Moment estimations of new Szász–Mirakyan–Durrmeyer operators, Appl. Math. Comput. 271 (2015), 540–547. 10.1016/j.amc.2015.09.037Search in Google Scholar

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

[13] G. C. Jain, Approximation of functions by a new class of linear operators, J. Aust. Math. Soc. 13 (1972), 271–276. 10.1017/S1446788700013689Search in Google Scholar

[14] J. P. King, Positive linear operators which preserve x2, Acta Math. Hungar. 99 (2003), no. 3, 203–208. 10.1023/A:1024571126455Search in 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. Search in 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. Search in 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. 10.1186/1029-242X-2013-586Search in 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. 10.1007/s40306-014-0077-9Search in Google Scholar

[19] N. Rao and A. Wafi, Stancu-variant of generalized Baskakov operators, Filomat 31 (2017), no. 9, 2625–2632. 10.2298/FIL1709625RSearch in Google 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. 10.1073/pnas.60.4.1196Search in Google Scholar

[21] O. Szasz, Generalization of S. Bernstein’s polynomials to the infinite interval, J. Res. Nat. Bur. Standards 45 (1950), 239–245. 10.6028/jres.045.024Search in Google Scholar

[22] S. Tarabie, On Jain-beta linear operators, Appl. Math. Inf. Sci. 6 (2012), no. 2, 213–216. Search in 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. 10.1501/Commua1_0000000240Search in Google Scholar

[24] Z. Ziegler, Linear approximation and generalized convexity, J. Approx. Theory 1 (1968), 420–443. 10.1016/0021-9045(68)90031-2Search in Google Scholar

Received: 2017-04-15
Revised: 2017-10-26
Accepted: 2017-11-08
Published Online: 2019-02-16
Published in Print: 2020-09-01

© 2020 Walter de Gruyter GmbH, Berlin/Boston

Articles in the same Issue

  1. Frontmatter
  2. Stability analysis of delay integro-differential equations of HIV-1 infection model
  3. Oscillation of first-order differential equations with several non-monotone retarded arguments
  4. Idempotent matrices with invertible transpose
  5. Some algebraic aspects of the gluing of differential spaces
  6. Inner structure in real vector spaces
  7. Averaged semi-discrete scheme of sum-approximation for one nonlinear multi-dimensional integro-differential parabolic equation
  8. Differential and integral equations for the 2-iterated Bernoulli, 2-iterated Euler and Bernoulli–Euler polynomials
  9. On adjoint resolutions and dimensions of modules
  10. Approximation by modified Jain–Baskakov operators
  11. Carleson measure and Volterra type operators on weighted BMOA spaces
  12. The effect of perturbations of frames and fusion frames on their redundancies
  13. Predual spaces of generalized grand Morrey spaces over non-doubling measure spaces
  14. Trigonometric identities inspired by the atomic form factor
  15. W(Lp,Lq) boundedness of localization operators associated with the Stockwell transform
  16. Voronovskaja’s theorem for functions with exponential growth
  17. A quantitative Balian–Low theorem for higher dimensions
  18. Lp(·)–Lq(·) boundedness of some integral operators obtained by extrapolation techniques
  19. Statistical convergence of multiple sequences on a product time scale
https://doi.org/10.1515/gmj-2019-2008
Keywords for this article
Jain operators; Baskakov operators; King type operators; weighted approximation

Articles in the same Issue

  1. Frontmatter
  2. Stability analysis of delay integro-differential equations of HIV-1 infection model
  3. Oscillation of first-order differential equations with several non-monotone retarded arguments
  4. Idempotent matrices with invertible transpose
  5. Some algebraic aspects of the gluing of differential spaces
  6. Inner structure in real vector spaces
  7. Averaged semi-discrete scheme of sum-approximation for one nonlinear multi-dimensional integro-differential parabolic equation
  8. Differential and integral equations for the 2-iterated Bernoulli, 2-iterated Euler and Bernoulli–Euler polynomials
  9. On adjoint resolutions and dimensions of modules
  10. Approximation by modified Jain–Baskakov operators
  11. Carleson measure and Volterra type operators on weighted BMOA spaces
  12. The effect of perturbations of frames and fusion frames on their redundancies
  13. Predual spaces of generalized grand Morrey spaces over non-doubling measure spaces
  14. Trigonometric identities inspired by the atomic form factor
  15. W(Lp,Lq) boundedness of localization operators associated with the Stockwell transform
  16. Voronovskaja’s theorem for functions with exponential growth
  17. A quantitative Balian–Low theorem for higher dimensions
  18. Lp(·)–Lq(·) boundedness of some integral operators obtained by extrapolation techniques
  19. Statistical convergence of multiple sequences on a product time scale
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