scholarly journals On convergence of certain nonlinear Bernstein operators

Filomat ◽  
2016 ◽  
Vol 30 (1) ◽  
pp. 141-155 ◽  
Author(s):  
Harun Karsli ◽  
Ismail Tiryaki ◽  
Erhan Altin
Keyword(s):  
Pointwise Convergence ◽  
Bernstein Polynomials ◽  
Bernstein Operators ◽  
Bounded Functions ◽  
Derivatives Of

In this article, we concern with the nonlinear Bernstein operators NBnf of the form (NBnf)(x)= n?k=0 Pn,k (x,f (k/n)), 0 ? x ? 1, n?N, acting on bounded functions on an interval [0,1], where Pn,k satisfy some suitable assumptions. As a continuation of the very recent paper of the authors [22], we estimate their pointwise convergence to a function f having derivatives of bounded (Jordan) variation on the interval [0,1]. We note that our results are strict extensions of the classical ones, namely, the results dealing with the linear Bernstein polynomials.

scholarly journals Some approximation properties of a certain nonlinear Bernstein operators

Filomat ◽  
2014 ◽  
Vol 28 (6) ◽  
pp. 1295-1305 ◽  
Author(s):  
Harun Karsli ◽  
Ismail Tiryaki ◽  
Erhan Altin
Keyword(s):  
Pointwise Convergence ◽  
Bernstein Polynomials ◽  
Bernstein Operators ◽  
Approximation Properties ◽  
Approximation Operators ◽  
Bernstein Approximation ◽  
Bounded Functions

The present paper concerns with a certain sequence of nonlinear Bernstein operators NBnf of the form (NBnf )(x) = ?nk=0 Pk,n (x,f (k/n)), 0 ? x ? 1, n ? N, acting on bounded functions on an interval [0, 1], where Pk, n satisfy some suitable assumptions. We will also investigate the pointwise convergence of this operators in some functional spaces. As a result, this study can be considered as an extension of the results dealing with the linear Bernstein Polynomials. As far as we know this kind of study is the first one on the nonlinear Bernstein approximation operators.

The Bernstein operators on any finite interval revisited

2020 ◽  
Vol 29 (1) ◽  
pp. 01-08
Author(s):  
DAN BARBOSU
Keyword(s):  
Simultaneous Approximation ◽  
Finite Interval ◽  
Bernstein Polynomials ◽  
Bernstein Operators ◽  
Approximation Properties

One studies simultaneous approximation properties of fundamental Bernstein polynomials involved in the construction of the mentioned operators.

Direct and converse results for q-Bernstein operators

2009 ◽  
Vol 52 (2) ◽  
pp. 339-349 ◽  
Author(s):  
Zoltán Finta
Keyword(s):  
Bernstein Polynomials ◽  
Modulus Of Smoothness ◽  
Second Order ◽  
Bernstein Operators ◽  
Approximation Theorems ◽  
Converse Theorems ◽  
Direct And Converse Results ◽  
Direct Approximation ◽  
Converse Results

AbstractDirect and converse theorems are established for the q-Bernstein polynomials introduced by G. M. Phillips. The direct approximation theorems are given for the second-order Ditzian–Totik modulus of smoothness. The converse results are theorems of Berens–Lorentz type.

scholarly journals On partial derivatives of multivariate Bernstein polynomials

2016 ◽  
Vol 26 (4) ◽  
pp. 294-305 ◽  
Author(s):  
A. Yu. Veretennikov ◽  
E. V. Veretennikova
Keyword(s):  
Bernstein Polynomials ◽  
Partial Derivatives ◽  
Derivatives Of

scholarly journals Lagrange-type operators associated with Uan

2014 ◽  
Vol 96 (110) ◽  
pp. 159-168 ◽  
Author(s):  
Heiner Gonska ◽  
Ioan Raşa ◽  
Elena-Dorina Stănilă
Keyword(s):  
Main Tool ◽  
Linear Operators ◽  
Positive Linear Operators ◽  
Bernstein Operators ◽  
Boolean Sums ◽  
Derivatives Of

We consider a class of positive linear operators which, among others, constitute a link between the classical Bernstein operators and the genuine Bernstein-Durrmeyer mappings. The focus is on their relation to certain Lagrange-type interpolators associated to them, a well known feature in the theory of Bernstein operators. Considerations concerning iterated Boolean sums and the derivatives of the operator images are included. Our main tool is the eigenstructure of the members of the class.

scholarly journals DERIVATIVES OF BERNSTEIN OPERATORS AND SMOOTHNESS WITH JACOBI WEIGHTS

2010 ◽  
Vol 14 (4) ◽  
pp. 1491-1500 ◽  
Author(s):  
Jianjun Wang ◽  
Zongben Xu ◽  
Guodong Han
Keyword(s):  
Bernstein Operators ◽  
Jacobi Weights ◽  
Derivatives Of

scholarly journals Better degree of approximation by modified Bernstein-Durrmeyer type operators

2021 ◽  
Vol 0 (0) ◽  
pp. 0
Author(s):  
Purshottam Narain Agrawal ◽  
Şule Yüksel Güngör ◽  
Abhishek Kumar
Keyword(s):  
Present Article ◽  
Degree Of Approximation ◽  
Modulus Of Smoothness ◽  
Bernstein Operators ◽  
Approximation Of Functions ◽  
Careful Choice ◽  
Durrmeyer Type Operators ◽  
Durrmeyer Variant ◽  
Derivatives Of ◽  
Infinitely Differentiable Function

<p style='text-indent:20px;'>In the present article we investigate a Durrmeyer variant of the generalized Bernstein-operators based on a function <inline-formula><tex-math id="M1">\begin{document}$ \tau(x), $\end{document}</tex-math></inline-formula> where <inline-formula><tex-math id="M2">\begin{document}$ \tau $\end{document}</tex-math></inline-formula> is infinitely differentiable function on <inline-formula><tex-math id="M3">\begin{document}$ [0, 1], \; \tau(0) = 0, \tau(1) = 1 $\end{document}</tex-math></inline-formula> and <inline-formula><tex-math id="M4">\begin{document}$ \tau^{\prime }(x)&gt;0, \;\forall\;\; x\in[0, 1]. $\end{document}</tex-math></inline-formula> We study the degree of approximation by means of the modulus of continuity and the Ditzian-Totik modulus of smoothness. A Voronovskaja type asymptotic theorem and the approximation of functions with derivatives of bounded variation are also studied. By means of a numerical example, finally we illustrate the convergence of these operators to certain functions through graphs and show a careful choice of the function <inline-formula><tex-math id="M5">\begin{document}$ \tau(x) $\end{document}</tex-math></inline-formula> leads to a better approximation than the generalized Bernstein-Durrmeyer type operators considered by Kajla and Acar [<xref ref-type="bibr" rid="b11">11</xref>].</p>

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