Article Contents

2024, Volume 7, Issue 3: 297-309. Doi: 10.3934/mfc.2023005

This issue Previous Article Lyapunov type inequalities for nonlinear fractional Hamiltonian systems in the frame of conformable derivatives Next Article Optimal investment strategy for the DC pension plan based on jump diffusion model and S-shaped utility

Approximation by modified Bernstein polynomials based on real parameters

1.
Department of Mathematics Indira Gandhi National Tribal University, Lalpur Amarkantak 484887, Madhya Pradesh, India
2.
Department of Applied Sciences and Humanities, Institute of Engineering and Technology, Dr. A.P.J. Abdul Kalam Technical University, Lucknow 226021 Uttar Pradesh, India

^*Corresponding author: Vishnu Narayan Mishra
^*Corresponding author: Vishnu Narayan Mishra

Received: May 2022

Revised: January 2023

Early access: February 2023

Published: August 2024

Abstract / Introduction Full Text(HTML) Related Papers Cited by

Abstract

In this paper, we introduce a modified Bernstein-type operators based on two real parameters and study its various approximation properties. We derive some direct results e.g. Voronovkaja type asymptotic theorem, an estimate of error in ordinary as well as in Ditzian Totik modulus of smoothness and an error estimate for functions belonging to the Lipschitz type space. Further, we examine the rate of approximation for a Kirov and Popova type generalization of these operators.

Keywords:

Mathematics Subject Classification: Primary: 41A10, Secondary: 41A25, 41A36.

Citation:

Full Text(HTML)

Related Papers

Cited by

References

[1]	A. M. Acu and P. N. Agrawal, Better approximation of functions by genuine Bernstein-Durrmeyer type operators, Carpathian J. Math., 35 (2019), 125-136.
[2]	A. M. Acu, I. C. Buscu and I. Rasa, Generalized Kantorovich modifications of positive linear operators, Math. Found. of Comput., 4 (2022), 1-9.
[3]	A. M. Acu, V. Gupta and G. Tachev, Better numerical approximation by durrmeyer type operators, Result Math., 74 (2019), Paper No. 90, 24 pp. doi: 10.1007/s00025-019-1019-6.
[4]	A. M. Acu and V. A. Radu, Approximation by certain operators Linking the $\alpha-$Bernstein and the Genuine $\alpha-$Bernstein-Durrmeyer Operators, In International Conf. on Recent Advances in Pure and Appl. Math. Springer, Singap., 306 (2018), 77-88. doi: 10.1007/978-981-15-1153-0_7.
[5]	S. N. Bernstein, Démonstration du théoréme de Weierstrass fondee sur le calculdes probabilités, Commun. Soc. Math. Kharkov, 13 (1912), 1-2.
[6]	J. Bustamante, Bernstein Operators and Their Properties, Birkh$\ddot a$user, Cham, 2017. doi: 10.1007/978-3-319-55402-0.
[7]	P. Cardaliaguet and G. Euvrard, Approximation of a function and its derivative with a neural network, Neural Netw., 5 (1992), 207-220. doi: 10.1016/S0893-6080(05)80020-6.
[8]	N. Çetin and V. A. Radu, Approximation by generalized Bernstein-Stancu operators, Turkish J. Math., 43 (2019), 2032-2048. doi: 10.3906/mat-1903-109.
[9]	F. Cucker and D.-X. Zhou, Learning Theory: An Approximation Theory Viewpoint, Cambridge University Press, 2007. doi: 10.1017/CBO9780511618796.
[10]	M. M. Derriennic, Sur l'approximation des fonctions intégrables par despolynômes de Bernstein modifiés, J. Approx. Theory, 31 (1981), 325-343. doi: 10.1016/0021-9045(81)90101-5.
[11]	Z. Ditzian and V. Totik, Moduli of Smoothness, Springer-Verlag, New York, 1987. doi: 10.1007/978-1-4612-4778-4.
[12]	O. Doǧru, M. A. Özarslan and F. Taşdelen, On positive operators involving a certain class of generating functions, Stud. Sci. Math. Hungar, 41 (2004), 415-429. doi: 10.1556/SScMath.41.2004.4.5.
[13]	J. L. Durrmeyer, Une Formule d'inversion de La Transformée de Laplace, Applications a La Theorie Des Moments, These de 3e cycle Faculté des Sciences de l'Université de Paris, 1967.
[14]	H. Esser, On pointwise convergence estimates for positive linear operators on $\mathcal{C}[a, b]$, Indag. Math., 38 (1976), 189-194.
[15]	H. Feng, S. Hou, L. Wei and D. Zhou, CNN models for readability of Chinese texts, Math. Found. Comput., 5 (2022), 351-362.
[16]	I. Gavrea and M. Ivan, An answer to a conjecture on Bernstein operators, J. Math. Anal. Appl., 390 (2012), 86-92. doi: 10.1016/j.jmaa.2012.01.021.
[17]	H. Gonska, On the degree of approximation in Voronovskajas theorem, Studia Univ. Babe AAŸ Bolyai Math, 52 (2007), 103-115.
[18]	H. H. Gonska, Two problems on best constants in direct estimates in "Problem Section of Proc. Edmonton Conf. Approximation Theory", Amer. Math. Soc., Providence, (1983), 194.
[19]	H. Gonska and I. Rasa, Asymptotic behavior of differentiated Bernstein polynomials, Mat. Vesnik, 61 (2009), 53-60.
[20]	H. Gonska and D. Zhou, On an extremal problem concerning Bernstein operators, Serdica Math. J., 21 (1995), 137-150.
[21]	X. Guo, L. Li and Q. Wu, Modeling interactive components by coordinate kernel polynomial models, Math. Found. of Comput., 3 (2020), 263.
[22]	V. Gupta and R. P. Agarwal, Convergence Estimates in Approximation Theory, Springer, Cham, 2014. doi: 10.1007/978-3-319-02765-4.
[23]	V. Gupta and G. Tachev, Approximation with Positive Linear Operators and Linear Combinations, Developments in Mathematics, 50. Springer, Cham, 2017. doi: 10.1007/978-3-319-58795-0.
[24]	V. Gupta, G. Tachev and A. M. Acu, Modified Kantorovich operators with better approximation properties, Numer. Algorithms, 81 (2019), 125-149. doi: 10.1007/s11075-018-0538-7.
[25]	S. Huang, Y. Feng and Q. Wu, Learning theory of minimum error entropy under weak moment conditions, Anal. Appl., 20 (2022), 121-139. doi: 10.1142/S0219530521500044.
[26]	G. Içöz, A Kantorovich variant of a new type Bernstein-Stancu polynomials, Appl. Math. Comput., 218 (2012), 8552-8560. doi: 10.1016/j.amc.2012.02.017.
[27]	D. P. Kacsó, Discrete Jackson-type operators via a Boolean sum approach, J. Comput. Anal. and Appl., 3 (2001), 399-413. doi: 10.1023/A:1012054425159.
[28]	H. Khosravian-Arab, M. Dehghan and M. R. Eslahchi, A new approach to improve the order of approximation of the Bernstein operators: Theory and applications, Numer. Algorithms, 77 (2018), 111-150. doi: 10.1007/s11075-017-0307-z.
[29]	G. H. Kirov, Approximation with Quasi-Splines, Translated from the Bulgarian. Institute of Physics, Bristol, 1992.
[30]	G. H. Kirov, A generalization of the Bernstein polynomials, Math. Balkanica (NS), 6 (1992), 147-153.
[31]	G. H. Kirov and L. Popova, A generalization of the linear positive operators, Math. Balkanica (NS), 7 (1993), 149-162.
[32]	P. P. Korovkin, Linear Operators and Approximation Theory, Hindustan Publishing Corp. (India), Delhi 1960
[33]	Y. LeCun, L. Bottou, Y. Bengio and P. Haffner, Gradient-based learning applied to document recognition, Proceedings IEEE, 86 (1998), 2278-2324. doi: 10.1109/5.726791.
[34]	A. Lupas, Die Folge der Betaoperatoren, Dissertation, Alexandru, 1972.
[35]	F. S. Lv and J. Fan, Optimal learning with Gaussians and correntropy loss, Anal. Appl., 19 (2021), 107-124. doi: 10.1142/S0219530519410124.
[36]	V. N. Mishra, K. Khatri, L. N. Mishra and Deepmala, Inverse result in simultaneous approximation by Baskakov-Durrmeyer-Stancu operators, J. Ineq. Appl., 2013 (2013), 11 pp. doi: 10.1186/1029-242X-2013-586.
[37]	M. A. Özarslan and H. Aktuĝlu, Local approximation for certain King type operators, Filomat, 27 (2013), 173-181. doi: 10.2298/FIL1301173O.
[38]	T. Popoviciu, Sur l'approximation des fonctions covexes dordre supérieur, Mathematica (Cluj), 10 (1934), 49-54.
[39]	T. Popoviciu, Sur l'approximation des fonctions continues dune variable réellepar des polynomes, Ann. Sci. Univ. Iasi, Sect. I, Math, 28 (1942), 208.
[40]	R. Pratap, The Family of $\lambda$-Bernstein-Durrmeyer operators based on certain parameters, Mathe. Found.of Comput., 5 (2022), 1-12.
[41]	S. Rahman, M. Mursaleen and A. M. Acu, Approximation properties of $\lambda$-Bernstein-Kantorovich operators with shifted knots, Math. Method. Appl. Sci., 42 (2019), 4042-4053. doi: 10.1002/mma.5632.
[42]	P. C. Sikkema, Der Wert einiger Konstanten in der theorie der approximation mit Bernstein polynomen, Numer. Math., 3 (1961), 107-116. doi: 10.1007/BF01386008.
[43]	T. A. Sinha, P. N. Agrawal and K. K. Singh, Inverse theorem for the iterates of modified Bernstein type polynomials, Stud. Univ. Babeş-Bolyai Math., 59 (2014), 331-350.
[44]	G. Tachev, The complete asymptotic expansion for Bernstein operators, J. Math. Anal. Appl, 385 (2012), 1179-1183. doi: 10.1016/j.jmaa.2011.07.042.
[45]	E. Voronovskaja, Détermination de la forme asymptotique d'approximation des fonctions par les polynômes de M. Bernstein, CR Acad. Sci. URSS, 79 (1932), 79-85.
[46]	F. Wang, D. Yu and B. Zhang, On approximation of Bernstein-Durrmeyer operators in movable interval, Filomat, 35 (2021), 1191-1203. doi: 10.2298/fil2104191w.