Linear combinations of two Bernstein polynomials

Gancho Tachev

doi:10.3934/mfc.2022061

Article Contents

2023, Volume 6, Issue 3: 586-590. Doi: 10.3934/mfc.2022061

This issue Previous Article Starlike functions associated with $ \tanh z $ and Bernardi integral operator Next Article Analytic function that map the unit disk into the inside of the lemniscate of Bernoulli

Linear combinations of two Bernstein polynomials

Gancho Tachev^1,

Department of Mathematics, University of Architecture, Sofia 1046, Bulgaria

Dedicated to the 60-th Anniversary of prof. Vijay Gupta

Received: October 2022

Revised: November 2022

Early access: December 27, 2022

Published online: December 27, 2022

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

Abstract

In the present note we construct a linear combination $ L_{kn} $ of two Bernstein polynomials $ B_n $ and $ B_{kn} $, for natural number $ k\geq 2 $. The new operator $ L_{kn} $ reproduces all algebraic polynomials of degree $ \leq 2 $. A direct pointwise estimate is proved which shows that the rate of approximation of $ f\in C[0, 1] $ such that $ \|\varphi f^{(3)}\|_{C[0, 1]}<\infty $ by $ L_{kn} $ is $ O\left(\frac{1}{n^{\frac{5}{2}}} \right) $, when $ x $ is close to the ends of the interval $ [0, 1] $.

Keywords:

Mathematics Subject Classification: 41A10, 41A17, 41A25, 41A30, 41A36.

Citation:

Full Text(HTML)

Related Papers

Cited by

References

[1]	A. Acu, I. Buscu and I. Rasa, Generalized kantorovich modifications of positive linear operators, Mathematical Foundations of Computing, 6 (2023), 54-62. doi: 10.3934/mfc.2021042.
[2]	S.N. Bernstein, Démonstration du théoréme de Weierstrass fondee sur le calcul desprobabilities, Commun. Kharkov Math. Soc., 13 (1912/1913), 1-2.
[3]	J. Bustamante, Bernstein Operators and Their Properties, Birkhäuser/Springer, Cham, 2017. doi: 10.1007/978-3-319-55402-0.
[4]	P. L. Butzer, Linear combinations of Bernstein polynomials, Can. J. Math., 5 (1953), 559-567. doi: 10.4153/CJM-1953-063-7.
[5]	R. A. DeVore and G. G. Lorentz, Constructive Approximation, Springer, Berlin, 1993.
[6]	Z. Ditzian and K. G. Ivanov, Strong converse inequalities, J. Anal. Math., 61 (1993), 61-111. doi: 10.1007/BF02788839.
[7]	Z. Ditzian and V. Totik, Moduli of Smoothness, Springer, New York, 1987. doi: 10.1007/978-1-4612-4778-4.
[8]	M. Felten, Direct and inverse estimates for Bernstein polynomials, Constr. Approx., 14 (1998), 459-468. doi: 10.1007/s003659900084.
[9]	H. Feng, S. Hou, L. Wei and D. Zhou, CNN models for readability of Chinese texts, Mathematical Foundations of Computing, 5 (2022), 351-362. doi: 10.3934/mfc.2022021.
[10]	V. Gupta and G. Tachev, Approximation with Positive Linear Operators and Linear Combinations, Developments in Mathematics, vol.50, Springer, Cham, 2017. doi: 10.1007/978-3-319-58795-0.
[11]	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.
[12]	H. Karsli, On approximation to discrete $q$-derivatives of functions via $q$-Bernstein-Schurer operators, Bull. Transilv. Univ. Braşov Ser. III. Math. Comput. Sci., 2 (2022), 83-98. doi: 10.31926/but.mif.
[13]	G. G. Lorentz, Bernstein Polynomials, Mathematical Expositions 8, University of Toronto Press, 1953.
[14]	F. S. Lv and L. Fan, Optimal learning with Gaussians and correntropy, Anal. Appl., 19 (2021), 107-124. doi: 10.1142/S0219530519410124.