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On Better Approximation of the Squared Bernstein Polynomials | ||
| Journal of University of Anbar for Pure Science | ||
| Article 11, Volume 16, Issue 2, 2022, Pages 92-98 PDF (760.71 K) | ||
| Document Type: Research Paper | ||
| DOI: 10.37652/juaps.2022.176478 | ||
| Authors | ||
| Rafah Katham* ; Ali Mohammad | ||
| Department of Mathematics, College of Education for Pure Sciences, University of Basrah, Basrah, Iraq | ||
| 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 | ||
| References | ||
|
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[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.
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[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. | ||
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