Advanced Search
Article Contents
Article Contents

A numerical comparative study of generalized Bernstein-Kantorovich operators

  • *Corresponding author

    *Corresponding author
Abstract Full Text(HTML) Figure(6) / Table(3) Related Papers Cited by
  • In this paper, a new generalization of the Bernstein-Kantorovich type operators involving multiple shape parameters is introduced. Certain Voronovskaja and Grüss-Voronovskaya type approximation results, statistical convergence and statistical rate of convergence of proposed operators are obtained by means of a regular summability matrix. Some illustrative graphics that demonstrate the convergence behavior, accuracy and consistency of the operators are given via Maple algorithms. The proposed operators are comprehensively compared with classical Bernstein, Bernstein-Kantorovich and other new modifications of Bernstein operators such as $ \lambda $-Bernstein, $ \lambda $-Bernstein-Kantorovich, $ \alpha $-Bernstein and $ \alpha $-Bernstein-Kantorovich operators.

    Mathematics Subject Classification: Primary: 41A10, 41A25; Secondary: 41A36, 26A16, 40C05.


    \begin{equation} \\ \end{equation}
  • 加载中
  • Figure 1.  Generalized Bernstein basis polynomial with $ p = 4 $ and $ p = 5 $

    Figure 2.  Generalized Bernstein basis polynomial with $ p = 6 $ and $ p = 8 $

    Figure 3.  Approximations of operators $ \mathcal{K}_{p}(\vartheta; z;\lambda) $ with different $ p $ values

    Figure 4.  Errors of approximation with different $ p $ values

    Figure 5.  $[![]!]

    Figure 6.  $[![]!]

    Table 1.  Comparison of certain Kantorovich operators via maximum errors of approximation for the function $ \vartheta_1(z) $

    $ p $ BK [22] $ \lambda $-BK [1] $ \alpha $-BK [37] G. B. K.
    4 0.587665e-6 0.555805e-6 0.584478e-6 0.555805e-6
    5 0.490495e-6 0.450425e-6 0.490010e-6 0.449155e-6
    6 0.421873e-6 0.381607e-6 0.422809e-6 0.375689e-6
    8 0.330590e-6 0.296430e-6 0.332454e-6 0.281839e-6
    10 0.272233e-6 0.244693e-6 0.274122e-6 0.224854e-6
    16 0.178427e-6 0.163579e-6 0.179697e-6 0.139245e-6
    20 0.145198e-6 0.134763e-6 0.146148e-6 0.110875e-6
    40 0.752695e-7 0.722972e-7 0.755868e-7 0.547628e-7
     | Show Table
    DownLoad: CSV

    Table 2.  Comparison of various operators via maximum errors of approximation for the function $ \vartheta_2(z) $

    $p$ Bernstein [3] $\lambda$-B [6] $\alpha$-B [7] BK [22] $\lambda$-BK [1] $\alpha$-BK [37] G. B. K.
    4 0.82355 0.94247 0.92642 0.78600 0.78600 0.78028 0.78600
    5 0.76350 0.84155 0.83258 0.67014 0.67014 0.66573 0.67014
    8 0.58677 0.61784 0.62170 0.49110 0.51367 0.51911 0.46399
    10 0.49794 0.51722 0.52434 0.45009 0.46447 0.47119 0.38486
    20 0.35516 0.35857 0.36421 0.33501 0.33783 0.34315 0.30638
    40 0.25152 0.25210 0.25468 0.24304 0.24354 0.24604 0.23436
     | Show Table
    DownLoad: CSV

    Table 3.  Comparison of operators via maximum errors for the function $ \vartheta_3(z) $

    $ p $ $ \left| \mathcal{K}_p(\vartheta_3;z;\lambda_k)-\vartheta_3(z) \right| $ $ \left|\mathcal{K}_p(\vartheta_3;z;\lambda_s)-\vartheta_3(z) \right| $ $ \left|B_p(\vartheta_3(z)) -\vartheta_3(z)\right| $
    $ 10 $ 0.595 0.585 0.639
    $ 20 $ 0.379 0.375 0.392
    $ 30 $ 0.275 0.275 0.282
     | Show Table
    DownLoad: CSV
  • [1] A. M. Acu, N. Manav and D. F. Sofonea, Approximation properties of $\lambda$-Kantorovich operators, J. Inequal. Appl., 2018 (2018), 12pp. doi: 10.1186/s13660-018-1795-7.
    [2] A. Alotaibi, F. Özger and S. A. Mohiuddine and M. A. Alghamdi et al, Approximation of functions by a class of Durrmeyer-Stancu type operators which includes Euler's beta function, Adv. Differ. Equ., 2021 (2021), 14pp. doi: 10.1186/s13662-020-03164-0.
    [3] S. N. Bernstein, Démonstration du théorème de Weierstrass fondée sur le calcul des probabilités, Comm. Soc. Math. Kharkow, 13 (1912), 1-2. 
    [4] H. Bohman, On approximation of continuous and of analytic functions, Ark. Math., 2 (1952), 43-56.  doi: 10.1007/BF02591381.
    [5] Q. B. Cai, The Bézier variant of Kantorovich type $\lambda$-Bernstein operators, J. Inequal. Appl., 2018 (2018), 10pp. doi: 10.1186/s13660-018-1688-9.
    [6] Q. B. Cai, B-Y. Lian and G. Zhou, Approximation properties of $\lambda$-Bernstein operators, J. Inequal. Appl., 2018 (2018), 11pp. doi: 10.1186/s13660-018-1653-7.
    [7] X. ChenJ. TanZ. Liu and J. Xie, Approximation of functions by a new family of generalized Bernstein operators, J. Math. Anal. Appl., 450 (2017), 244-261.  doi: 10.1016/j.jmaa.2016.12.075.
    [8] F. Dirik and K. Demirci, Korovkin type approximation theorems in B-statistical sense, Math. Comput. Modelling, 49 (2009), 2037-2044.  doi: 10.1016/j.mcm.2008.11.002.
    [9] Z. Ditzian and V. Totik, Moduli of Smoothness, Springer-Verlag, New York, 1987. doi: 10.1007/978-1-4612-4778-4.
    [10] R. A. DeVore and G. G. Lorentz, Constructive Approximation, Grundlehren der Mathematischen Wissenschaften [Fundamental Principles of Mathematical Sciences], 303. Springer-Verlag, Berlin, 1993. doi: 10.1007/978-3-662-02888-9.
    [11] O. Duman and C. Orhan, Statistical approximation by positive linear operators, Studia Math., 161 (2004), 187-197.  doi: 10.4064/sm161-2-6.
    [12] M. S. ErdoğanÇ. Dişibüyük and Ö. E. Oruç, An alternative distribution function estimation method using rational Bernstein polynomials, J. Comput. Appl. Math., 353 (2019), 232-242.  doi: 10.1016/j.cam.2018.12.033.
    [13] Z. Finta, Remark on Voronovskaja theorem for q-Bernstein operators, Stud. Univ. Babes-Bolyai Math., 56 (2011), 335-339. 
    [14] A. D. Gadjiev and C. Orhan, Some approximation theorems via statistical convergence, Rocky Mountain J. Math., 32 (2002), 129-138.  doi: 10.1216/rmjm/1030539612.
    [15] S. G. Gal and H. Gonska, Grüss and Grüss-Voronovskaya-type estimates for some Bernstein-type polynomials of real and complex variables, Jaen J. Approx., 7 (2015), 97-122. 
    [16] G. Grüss, Uber das Maximum des absoluten Betrages von $\frac{1}{b-a}\int_{a}^{b} \vartheta(x)g(x)dx -\frac{1}{(b-a)^2}\int_{a}^{b} \vartheta(x)dx\int_{a}^{b} g (x)dx$, Math. Z., 39 (1935), 215-226.  doi: 10.1007/BF01201355.
    [17] X. HanY. C. Ma and X. L. Huang, A novel generalization of Bézier curve and surface, J. Comput. Appl. Math., 217 (2008), 180-193.  doi: 10.1016/j.cam.2007.06.027.
    [18] G. Hu, C. Bo and X. Qin, Continuity conditions for Q-Bézier curves of degree n, J. Inequal. Appl., 2017 (2017), 14pp. doi: 10.1186/s13660-017-1390-3.
    [19] A. Il'inskii and S. Ostrovska, Convergence of generalized Bernstein polynomials, J. Approx. Theory, 116 (2002), 100-112.  doi: 10.1006/jath.2001.3657.
    [20] U. Kadak, On relative weighted summability in modular function spaces and associated approximation theorems, Positivity, 21 (2017), 1593-1614.  doi: 10.1007/s11117-017-0487-8.
    [21] U. KadakV. N. Mishra and S. Pandey, Chlodowsky type generalization of (p, q)-Szász operators involving Brenke type polynomials, Rev. R. Acad. Cienc. Exactas F¨ªs. Nat. Ser. A Mat. RACSAM, 112 (2018), 1443-1462.  doi: 10.1007/s13398-017-0439-y.
    [22] L. Kantorovich, Sur certains développements suivant les polyn\^{o}mes de la forme de S. Bernstein, Ⅰ, Ⅱ, C. R. Acad. Sci. URSS, (1930), 595–600.
    [23] H. Khosravian-ArabM. Dehghan and M. R. Eslahchi, A new approach to improve the order of approximation of the Bernstein operators: Theory and applications, Numer. Algor., 77 (2018), 111-150.  doi: 10.1007/s11075-017-0307-z.
    [24] P. P. Korovkin, On convergence of linear positive operators in the space of continuous functions, Dokl. Akad. Nauk. SSSR, 90 (1953), 961-964. 
    [25] U. Kadak, Generalized statistical convergence based on fractional order difference operator and its applications to approximation theorems, Iran. J. Sci. Technol. Trans. A Sci., 43 (2019), 225-237.  doi: 10.1007/s40995-017-0400-0.
    [26] U. Kadak, Generalized weighted invariant mean based on fractional difference operato r with applications to approximation theorems for functions of two variables, Results Math., 72 (2017), 1181-1202.  doi: 10.1007/s00025-016-0634-8.
    [27] U. Kadak, Modularly weighted four dimensional matrix summability with application to Korovkin type approximation theorem, J. Math. Anal. Appl., 468 (2018), 38-63.  doi: 10.1016/j.jmaa.2018.06.047.
    [28] U. KadakN. L. Braha and H. M. Srivastava, Statistical weighted B-summability and its applications to approximation theorems, Appl. Math. Comput., 302 (2017), 80-96.  doi: 10.1016/j.amc.2017.01.011.
    [29] U. Kadak and M. Ŏzlŭk, Extended Bernstein-Kantorovich-Stancu operators with multiple parameters and approximation properties, Numer. Funct. Anal. Optim., 42 (2021), 523-550.  doi: 10.1080/01630563.2021.1895833.
    [30] K. Kanat and M. Sofyalioğlu, Some approximation results for Stancu type Lupaş-Schurer operators based on (p, q)-sntegers, Appl. Math. Comput., 317 (2018), 129-142.  doi: 10.1016/j.amc.2017.08.046.
    [31] A. KajlaS. A. Mohiuddine and A. Alotaibi, Blending-type approximation by Lupaş-Durrmeyer-type operators involving Pólya distribution, Math. Meth. Appl. Sci., 44 (2021), 9407-9418.  doi: 10.1002/mma.7368.
    [32] I. KucukogluB. Simsek and Y. Simsek, Multidimensional Bernstein polynomials and Bezier curves: Analysis of machine learning algorithm for facial expression recognition based on curvature, Appl. Math. Comput., 344–345 (2019), 150-162.  doi: 10.1016/j.amc.2018.10.012.
    [33] S. A. MohiuddineN. AhmadF. ÖzgerA. Alotaibi and et al, Approximation by the parametric generalization of Baskakov-Kantorovich operators linking with Stancu operators, Iran J. Sci. Technol. Trans. A Sci., 45 (2021), 593-605.  doi: 10.1007/s40995-020-01024-w.
    [34] S. A. Mohiuddine and F. Özger, Approximation of functions by Stancu variant of Bernstein-Kantorovich operators based on shape parameter $\alpha$, Rev. R. Acad. Cienc. Exactas Fís. Nat. Ser. A Math. RACSAM, 114 (2020), 17pp. doi: 10.1007/s13398-020-00802-w.
    [35] S. A. Mohiuddine, Approximation by bivariate generalized Bernstein-Schurer operators and associated GBS operators, Adv. Difference Equ., 2020 (2020), 17pp. doi: 10.1186/s13662-020-03125-7.
    [36] S. A. Mohiuddine and B. A. S. Alamri, Generalization of equi-statistical convergence via weighted lacunary sequence with associated Korovkin and Voronovskaya type approximation theorems, Rev. R. Acad. Cienc. Exactas Fís. Nat. Ser. A Mat. RACSAM, 113 (2019), 1955-1973.  doi: 10.1007/s13398-018-0591-z.
    [37] S. A. MohiuddineT. Acar and A. Alotaibi, Construction of a new family of Bernstein-Kantorovich operators, Math. Meth. Appl. Sci., 40 (2017), 7749-7759.  doi: 10.1002/mma.4559.
    [38] M. A. Ŏzarslan and H. Aktuğlu, Local approximation properties for certain King type operators, Filomat, 27 (2013), 173-181.  doi: 10.2298/FIL1301173O.
    [39] F. Özger, Weighted statistical approximation properties of univariate and bivariate $\lambda$-Kantorovich operators, Filomat, 33 (2019), 3473-3486.  doi: 10.2298/FIL1911473O.
    [40] F. Özger, K. Demirci and S. Yıldız, Approximation by kantorovich variant of $\lambda$-schurer operators and related numerical results, Topics in Contemporary Mathematical Analysis and Applications, Boca Raton, USA: CRC Press, (2020), 77–94.
    [41] F. Özger, On new Bézier bases with Schurer polynomials and corresponding results in approximation theory, Commun. Fac. Sci. Univ. Ank. Ser. A1 Math. Stat., 69 (2020), 376-393.  doi: 10.31801/cfsuasmas.510382.
    [42] F. Özger, Applications of generalized weighted statistical convergence to approximation theorems for functions of one and two variables, Numer. Funct. Anal. Optim., 41 (2020), 1990-2006.  doi: 10.1080/01630563.2020.1868503.
    [43] F. Özger, H. M. Srivastava and S. A. Mohiuddine, Approximation of functions by a new class of generalized Bernstein-Schurer operators, Rev. R. Acad. Cienc. Exactas Fís. Nat. Ser. A Math. RACSAM, 114 (2020), 21pp. doi: 10.1007/s13398-020-00903-6.
    [44] H. M. Srivastava, F. Özger and S. A. Mohiuddine, Construction of Stancu-type Bernstein operators based on Bézier bases with shape parameter $\lambda$, phSymmetry, 11 (2019). doi: 10.3390/symxx010005.
    [45] H. M. SrivastavaK. J. AnsariF. Özger and Z. Ödemiş Özger, A link between approximation theory and summability methods via four-dimensional infinite matrices, Mathematics, 9 (2021), 1895.  doi: 10.3390/math9161895.
    [46] H. R. Tabrizidooz and K. Shabanpanah, Bernstein polynomial basis for numerical solution of boundary value problems, Numer. Algor., 77 (2018), 211-228.  doi: 10.1007/s11075-017-0311-3.
    [47] V. K. Weierstrass, Ueber die analytische Darstellbarkeit sogennanter willkürlicher Functionen einer reellen Veranderlichep, sp:, Sitzungsberichte der Akademie zu Berlin, (1885), 633–639.
    [48] J. X. Xiang, Expansion of moments of Bernstein polynomials, J. Math. Anal. Appl., 476 (2019), 585-594.  doi: 10.1016/j.jmaa.2019.03.072.
  • 加载中




Article Metrics

HTML views(651) PDF downloads(300) Cited by(0)

Access History

Other Articles By Authors



    DownLoad:  Full-Size Img  PowerPoint