November  2021, 4(4): 311-332. doi: 10.3934/mfc.2021021

A numerical comparative study of generalized Bernstein-Kantorovich operators

1. 

Department of Mathematics, Gazi University, Ankara-06100, Turkey, Yaşamkent Street, Bolu-14100, Turkey

2. 

Department of Engineering Sciences, İzmir Katip Çelebi University, İzmir-35620, Turkey

*Corresponding author

Received  June 2021 Revised  August 2021 Published  November 2021 Early access  September 2021

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.

Citation: Uğur Kadak, Faruk Özger. A numerical comparative study of generalized Bernstein-Kantorovich operators. Mathematical Foundations of Computing, 2021, 4 (4) : 311-332. doi: 10.3934/mfc.2021021
References:
[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.  Google Scholar

[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.  Google Scholar

[3]

S. N. Bernstein, Démonstration du théorème de Weierstrass fondée sur le calcul des probabilités, Comm. Soc. Math. Kharkow, 13 (1912), 1-2.   Google Scholar

[4]

H. Bohman, On approximation of continuous and of analytic functions, Ark. Math., 2 (1952), 43-56.  doi: 10.1007/BF02591381.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[9]

Z. Ditzian and V. Totik, Moduli of Smoothness, Springer-Verlag, New York, 1987. doi: 10.1007/978-1-4612-4778-4.  Google Scholar

[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.  Google Scholar

[11]

O. Duman and C. Orhan, Statistical approximation by positive linear operators, Studia Math., 161 (2004), 187-197.  doi: 10.4064/sm161-2-6.  Google Scholar

[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.  Google Scholar

[13]

Z. Finta, Remark on Voronovskaja theorem for q-Bernstein operators, Stud. Univ. Babes-Bolyai Math., 56 (2011), 335-339.   Google Scholar

[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.  Google Scholar

[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.   Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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. Google Scholar

[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.  Google Scholar

[24]

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

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[38]

M. A. Ŏzarslan and H. Aktuğlu, Local approximation properties for certain King type operators, Filomat, 27 (2013), 173-181.  doi: 10.2298/FIL1301173O.  Google Scholar

[39]

F. Özger, Weighted statistical approximation properties of univariate and bivariate $\lambda$-Kantorovich operators, Filomat, 33 (2019), 3473-3486.  doi: 10.2298/FIL1911473O.  Google Scholar

[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. Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[47]

V. K. Weierstrass, Ueber die analytische Darstellbarkeit sogennanter willkürlicher Functionen einer reellen Veranderlichep, sp:, Sitzungsberichte der Akademie zu Berlin, (1885), 633–639. Google Scholar

[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.  Google Scholar

show all references

References:
[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.  Google Scholar

[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.  Google Scholar

[3]

S. N. Bernstein, Démonstration du théorème de Weierstrass fondée sur le calcul des probabilités, Comm. Soc. Math. Kharkow, 13 (1912), 1-2.   Google Scholar

[4]

H. Bohman, On approximation of continuous and of analytic functions, Ark. Math., 2 (1952), 43-56.  doi: 10.1007/BF02591381.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[9]

Z. Ditzian and V. Totik, Moduli of Smoothness, Springer-Verlag, New York, 1987. doi: 10.1007/978-1-4612-4778-4.  Google Scholar

[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.  Google Scholar

[11]

O. Duman and C. Orhan, Statistical approximation by positive linear operators, Studia Math., 161 (2004), 187-197.  doi: 10.4064/sm161-2-6.  Google Scholar

[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.  Google Scholar

[13]

Z. Finta, Remark on Voronovskaja theorem for q-Bernstein operators, Stud. Univ. Babes-Bolyai Math., 56 (2011), 335-339.   Google Scholar

[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.  Google Scholar

[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.   Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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. Google Scholar

[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.  Google Scholar

[24]

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

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[38]

M. A. Ŏzarslan and H. Aktuğlu, Local approximation properties for certain King type operators, Filomat, 27 (2013), 173-181.  doi: 10.2298/FIL1301173O.  Google Scholar

[39]

F. Özger, Weighted statistical approximation properties of univariate and bivariate $\lambda$-Kantorovich operators, Filomat, 33 (2019), 3473-3486.  doi: 10.2298/FIL1911473O.  Google Scholar

[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. Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[47]

V. K. Weierstrass, Ueber die analytische Darstellbarkeit sogennanter willkürlicher Functionen einer reellen Veranderlichep, sp:, Sitzungsberichte der Akademie zu Berlin, (1885), 633–639. Google Scholar

[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.  Google Scholar

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.  Approximations of Generalized Bernstein-Kantorovich operators for different values of $ p $
Figure 6.  Errors of approximation for the function $ \vartheta_4(z) $
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
$ 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
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
$p$ 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
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
$ 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
[1]

M. Delgado Pineda, E. A. Galperin, P. Jiménez Guerra. MAPLE code of the cubic algorithm for multiobjective optimization with box constraints. Numerical Algebra, Control & Optimization, 2013, 3 (3) : 407-424. doi: 10.3934/naco.2013.3.407

[2]

Danilo Costarelli, Gianluca Vinti. Asymptotic expansions and Voronovskaja type theorems for the multivariate neural network operators. Mathematical Foundations of Computing, 2020, 3 (1) : 41-50. doi: 10.3934/mfc.2020004

[3]

David Cheban, Cristiana Mammana. Continuous dependence of attractors on parameters of non-autonomous dynamical systems and infinite iterated function systems. Discrete & Continuous Dynamical Systems, 2007, 18 (2&3) : 499-515. doi: 10.3934/dcds.2007.18.499

[4]

De-Jun Feng, Antti Käenmäki. Equilibrium states of the pressure function for products of matrices. Discrete & Continuous Dynamical Systems, 2011, 30 (3) : 699-708. doi: 10.3934/dcds.2011.30.699

[5]

Aleksander Ćwiszewski, Wojciech Kryszewski. On a generalized Poincaré-Hopf formula in infinite dimensions. Discrete & Continuous Dynamical Systems, 2011, 29 (3) : 953-978. doi: 10.3934/dcds.2011.29.953

[6]

Patrick Foulon, Boris Hasselblatt. Lipschitz continuous invariant forms for algebraic Anosov systems. Journal of Modern Dynamics, 2010, 4 (3) : 571-584. doi: 10.3934/jmd.2010.4.571

[7]

Kaizhi Wang, Jun Yan. Lipschitz dependence of viscosity solutions of Hamilton-Jacobi equations with respect to the parameter. Discrete & Continuous Dynamical Systems, 2016, 36 (3) : 1649-1659. doi: 10.3934/dcds.2016.36.1649

[8]

Pavel Krejčí, Thomas Roche. Lipschitz continuous data dependence of sweeping processes in BV spaces. Discrete & Continuous Dynamical Systems - B, 2011, 15 (3) : 637-650. doi: 10.3934/dcdsb.2011.15.637

[9]

Yong Fang, Patrick Foulon, Boris Hasselblatt. Longitudinal foliation rigidity and Lipschitz-continuous invariant forms for hyperbolic flows. Electronic Research Announcements, 2010, 17: 80-89. doi: 10.3934/era.2010.17.80

[10]

Mitsuharu Ôtani, Yoshie Sugiyama. Lipschitz continuous solutions of some doubly nonlinear parabolic equations. Discrete & Continuous Dynamical Systems, 2002, 8 (3) : 647-670. doi: 10.3934/dcds.2002.8.647

[11]

Mohsen Mousavi, Ali Zaghian, Morteza Esmaeili. Involutory-Multiple-Lightweight MDS Matrices based on Cauchy-type Matrices. Advances in Mathematics of Communications, 2021, 15 (4) : 589-610. doi: 10.3934/amc.2020084

[12]

Anwa Zhou, Jinyan Fan. A semidefinite relaxation algorithm for checking completely positive separable matrices. Journal of Industrial & Management Optimization, 2019, 15 (2) : 893-908. doi: 10.3934/jimo.2018076

[13]

Christian Budde, Marjeta Kramar Fijavž. Bi-Continuous semigroups for flows on infinite networks. Networks & Heterogeneous Media, 2021, 16 (4) : 553-567. doi: 10.3934/nhm.2021017

[14]

Piernicola Bettiol, Hélène Frankowska. Lipschitz regularity of solution map of control systems with multiple state constraints. Discrete & Continuous Dynamical Systems, 2012, 32 (1) : 1-26. doi: 10.3934/dcds.2012.32.1

[15]

Jianjun Liu, Min Zeng, Yifan Ge, Changzhi Wu, Xiangyu Wang. Improved Cuckoo Search algorithm for numerical function optimization. Journal of Industrial & Management Optimization, 2020, 16 (1) : 103-115. doi: 10.3934/jimo.2018142

[16]

Débora A. F. Albanez, Maicon J. Benvenutti. Continuous data assimilation algorithm for simplified Bardina model. Evolution Equations & Control Theory, 2018, 7 (1) : 33-52. doi: 10.3934/eect.2018002

[17]

Harry L. Johnson, David Russell. Transfer function approach to output specification in certain linear distributed parameter systems. Conference Publications, 2003, 2003 (Special) : 449-458. doi: 10.3934/proc.2003.2003.449

[18]

Giselle A. Monteiro, Milan Tvrdý. Generalized linear differential equations in a Banach space: Continuous dependence on a parameter. Discrete & Continuous Dynamical Systems, 2013, 33 (1) : 283-303. doi: 10.3934/dcds.2013.33.283

[19]

Rashad M. Asharabi, Jürgen Prestin. Computing eigenpairs of two-parameter Sturm-Liouville systems using the bivariate sinc-Gauss formula. Communications on Pure & Applied Analysis, 2020, 19 (8) : 4143-4158. doi: 10.3934/cpaa.2020185

[20]

François Hamel, Régis Monneau, Jean-Michel Roquejoffre. Asymptotic properties and classification of bistable fronts with Lipschitz level sets. Discrete & Continuous Dynamical Systems, 2006, 14 (1) : 75-92. doi: 10.3934/dcds.2006.14.75

 Impact Factor: 

Article outline

Figures and Tables

[Back to Top]