On Better Approximation of the Squared Bernstein Polynomials

Katham, Rafah; Mohammad, Ali

doi:10.37652/juaps.2022.176478

Print ISSN: 1991-8941

Online ISSN: 2706-6703

Volume16, Issue2

On Better Approximation of the Squared Bernstein Polynomials

Rafah Katham

Ali Mohammad

Journal of University of Anbar for Pure Science, 2022, Volume 16, Issue 2, Pages 92-98
10.37652/juaps.2022.176478

Abstract

The present paper is defined a new better approximation of the squared Bernstein polynomials. This better approximation has been built on a positive function defined on the interval [0,1] which has some properties. First, the moderate uniform convergence theorem for a sequence of linear positive operators (the generalization of the Korovkin theorem) of these polynomials is improved. Then, the rate of convergence of these polynomials corresponding to the first and second modulus of continuity and Ditzian- Totik modulus of smoothness is given. Also, the quantitative Voronovskaja and the Grüss- Voronovskaja theorems are discussed. Finally, some numerically applied for these polynomials are given by choosing a test function and two different functions show the effect of the different chosen functions . It turns the new better approximation of the squared Bernstein polynomials gives us a better numerical result than the numerical results of both the classical Bernstein polynomials and the squared Bernstein polynomials. MSC 2010. 41A10, 41A25, 41A36.

Keywords:

Squared B ernstein polynomial

Modulus of Smoothness

Quantitative Voronovskaja theorem

and Grüss-Voronovskaja theorem

Main Subjects:

Mathematical

PDF (566 K)

[1] J.P. King, Positive linear operators which preserve x2, Acta Math. Hungar, 99 (3) (2003) 203–208.

[2] D. Cárdenas-Morales a, P. Garranchoa, I. Raşa b, Bernstein-type operators which preserve polynomials. Computers and Mathematics with Applications 62 (2011) 158–163.

[3] I. Gavrea, M. Ivan, On a new sequence of positive linear operators related to squared Bernstein polynomials, Positivity. 21 (2017) 911–917.

[4] I. A. Abdul Samad, Ali J. Mohammad, The -th Powers of the Rational Bernstein Polynomials.Basrah Journal of Science 39(2)(2021) 179-191.

[5] Ali Jassim Mohammad, Asma Jaber, On Some Approximation Properties for a Sequence of -Bernstein Type Operators. Iraqi Journal of Science, 2021, Vol. 62, No. 5, pp: 1666-1674.

[6] Ali J.Mohammad , Amal K.Hassan, Simultaneous Approximation by a New Sequence of Integral Type Based on Two Parameters. Iraqi Journal of Science, 2021, Vol. 62, No. 5, pp: 1666-1674.

[7] S.A. Mohiuddine, T. Acar, M.G. Alghamdi, Genuine modiﬁed Bernstein-Durrmeyer operators. J. Inequal. Appl. 104 (2018) 1-10.

[8] SG. Gal, H Gonska, Grüss and Grüss-Voronovskaya-type estimates for some Bernstein-type polynomials of real and complex variables. Jaen J Approx.7(1) (2015) 97-122.

Article View: 33
PDF Download: 37