Convergence of rational Bernstein operators

Render, Hermann

doi:10.1016/j.amc.2014.01.152

Cited by 7 publications

(5 citation statements)

References 22 publications

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“…The convergence of R n;m is established under Q nÀ1;mÀ1 ðx; yÞ ¼ B nÀ1;mÀ1 uðx; yÞ ð31Þ for a given function u 2 CðDÞ. Recently, H. Render (see [10]) removed the special requirement (6) in the univariate case and formulated convergence results under the assumption that D n ¼ sup i¼0;...;nÀ1…”

Section: Discussionmentioning

confidence: 99%

“…Bernstein in [13] for proving the Weierstrass Approximation Theorem. Bernstein operators have been investigated by many mathematicians, see [1][2][3][4][5]9,10,12], and they have several applications in Computer Aided Geometric Design, see [1,6,11,12]. For the history of theory and applications of Bernstein operators, we refer to Farouki's paper [12].…”

Section: Introductionmentioning

confidence: 99%

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A family of bivariate rational Bernstein operators

Zhu

Xia

2015

Applied Mathematics and Computation

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Section: Discussionmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

A family of bivariate rational Bernstein operators

Zhu

Xia

2015

Applied Mathematics and Computation

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“…Remark: Before giving the proofs of the lemmas we would like to illustrate some functions (Usta, and Betus, 2020;Ait-Haddou and Mazure, 2016;Render, 2014;Usta, 2021;Usta, 2020;Yilmaz et.al., 2020) which are helpful for the proof of our lemmas. The functions Since 𝑣!…”

Section: Methodsmentioning

confidence: 99%

Estimation Features by Transformed Bernstein kind Polynomials

Mishra,

Garg,

Sharma

et al. 2023

IJERR

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In our extensive study of literature, we delved into the multifarious manifestations of discrete operator transformations. These transformations are pivotal in mathematical analysis, especially concerning Lebesgue integral equations. Our investigation led us to corroborate the findings of Acu, Heilmann and Lorentz particularly in the context of functions normed under the L1-norm. Generalization was a key facet of our research, wherein we probed deeper into these operators' behaviors. This endeavor yielded a profound result: the derivation of a global asymptotic formula, providing invaluable insight into the long-term trends exhibited by these operators. Such formulae are instrumental in predicting the operators' behaviors over an extended span. Furthermore, our exploration unveiled a plethora of findings related to these generalized operators. We meticulously computed various moments, shedding light on the statistical characteristics of these transformations. This included an investigation into convergence properties, essential for understanding the stability and reliability of the operators in question. One of the most noteworthy contributions of our study is the elucidation of pointwise approximation and direct results. These findings offer practical applications, allowing for precise and efficient approximations in practical scenarios. This is particularly significant in fields where these operators are routinely employed, such as signal processing, numerical analysis, and scientific computing. In essence, our research has not only confirmed the foundational work of Acu, Heilmann and Lorentz but has also expanded the horizons of knowledge surrounding discrete operator transformations, offering a wealth of insights and practical implications for a wide range of mathematical and computational applications.

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“…Please see [8] and [5]. In 2014, Render has discussed some convergence properties and error estimates of rational Bernstein polynomials in the general case [7].In 2017, Gavrea and Ivan have defined the square Bernstein polynomials as [3]:…”

Section: Introductionmentioning

confidence: 99%

The -th Powers of the Rational Bernstein Polynomials

Mohammad

2021

basjs

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This paper is defined and studied a new 𝑟-th powers of rational Bernstein polynomials. The convergence theorem, the recurrence relation for the 𝑚-th order moment, and the Voronovskaya-type asymptotic formula for these polynomials in ordinary approximation are given. Also, a numerical example for these polynomials is applied to approximate the test function sin 10𝑥 ∈ 𝐶,0,1-. The results obtained from this example are shown that these polynomials are given better than the corresponding numerical results for the classical Bernstein polynomials and the square rational Bernstein polynomials. The comparison is done by plot the graphs of the function and its approximations as well as the evaluation of the average absolute errors for these approximations.Article inf.

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Convergence of rational Bernstein operators

Cited by 7 publications

References 22 publications

A family of bivariate rational Bernstein operators

A family of bivariate rational Bernstein operators

Estimation Features by Transformed Bernstein kind Polynomials

The -th Powers of the Rational Bernstein Polynomials

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