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.

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.

scholarly journals Approximation by Jakimovski-Leviatan-Stancu-Durrmeyer type operators

Filomat ◽  
2019 ◽  
Vol 33 (6) ◽  
pp. 1517-1530 ◽  
Author(s):  
M. Mursaleen ◽  
Shagufta Rahman ◽  
Khursheed Ansari
Keyword(s):  
Upper Bound ◽  
Simultaneous Approximation ◽  
Approximation Theorem ◽  
Approximation Properties ◽  
Statistical Approximation ◽  
Durrmeyer Operators ◽  
Durrmeyer Type Operators

In the present paper, we introduce Stancu type modification of Jakimovski-Leviatan-Durrmeyer operators. First, we estimate moments of these operators. Next, we study the problem of simultaneous approximation by these operators. An upper bound for the approximation to rth derivative of a function by these operators is established. Furthermore, we obtain A-statistical approximation properties of these operators with the help of universal korovkin type statistical approximation theorem.

Voronovskaja’s theorem, shape preserving properties and iterations for complex q-Bernstein polynomials

2011 ◽  
Vol 48 (1) ◽  
pp. 23-43 ◽  
Author(s):  
Sorin Gal
Keyword(s):  
Convergence Theorem ◽  
Analytic Functions ◽  
Quantitative Estimate ◽  
Bernstein Polynomials ◽  
Exact Order ◽  
Approximation Properties ◽  
Shape Preserving ◽  
Voronovskaja’S Theorem ◽  
Shape Preserving Properties ◽  
Compact Disks

In this paper, first we prove Voronovskaja’s convergence theorem for complex q-Bernstein polynomials, 0 < q < 1, attached to analytic functions in compact disks in ℂ centered at origin, with quantitative estimate of this convergence. As an application, we obtain the exact order in approximation of analytic functions by the complex q-Bernstein polynomials on compact disks. Finally, we study the approximation properties of their iterates for any q > 0 and we prove that the complex qn-Bernstein polynomials with 0 < qn < 1 and qn → 1, preserve in the unit disk (beginning with an index) the starlikeness, convexity and spiral-likeness.

scholarly journals Approximation of functions by a new class of linear operators

1972 ◽  
Vol 13 (3) ◽  
pp. 271-276 ◽  
Author(s):  
G. C. Jain
Keyword(s):  
Negative Binomial ◽  
Bernstein Polynomials ◽  
Degree Of Approximation ◽  
Linear Operators ◽  
Approximation Properties ◽  
Convergence Properties ◽  
Approximation Of Functions ◽  
New Class ◽  
Binomial Distributions ◽  
Poisson Type

Various extensions and generalizations of Bernstein polynomials have been considered among others by Szasz [13], Meyer-Konig and Zeller [8], Cheney and Sharma [1], Jakimovski and Leviatan [4], Stancu [12], Pethe and Jain [11]. Bernstein polynomials are based on binomial and negative binomial distributions. Szasz and Mirakyan [9] have defined another operator with the help of the Poisson distribution. The operator has approximation properties similar to those of Bernstein operators. Meir and Sharma [7] and Jam and Pethe [3] deal with generalizations of Szasz-Mirakyan operator. As another generalization, we define in this paper a new operator with the help of a Poisson type distribution, consider its convergence properties and give its degree of approximation. The results for the Szasz-Mirakyan operator can easily be obtained from our operator as a particular case.

scholarly journals Exponential moments and simultaneous approximation properties for Durrmeyer type operators with weights of Szasz basis functions

Filomat ◽  
2021 ◽  
Vol 35 (5) ◽  
pp. 1465-1475
Author(s):  
Antonio-Jesús López-Moreno ◽  
Vijay Gupta
Keyword(s):  
Exponential Type ◽  
Exponential Growth ◽  
Simultaneous Approximation ◽  
Basis Functions ◽  
Approximation Properties ◽  
Exponential Functions ◽  
Quantitative Estimates ◽  
Exponential Moments ◽  
Durrmeyer Type Operators ◽  
Derivatives Of

The present paper deals with the approximation properties for exponential functions of general Durrmeyer type operators having the weights of Sz?sz basis functions. Here we give explicit expressions for exponential type moments by means of which we establish, for the derivatives of the operators, the Voronovskaja formulas for functions of exponential growth and the corresponding weighted quantitative estimates for the remainder in simultaneous approximation.

scholarly journals Some Approximation Properties for Modified Baskakov Type Operators

2005 ◽  
Vol 12 (2) ◽  
pp. 217-228
Author(s):  
Vijay Gupta
Keyword(s):  
Modulus Of Continuity ◽  
Simultaneous Approximation ◽  
Higher Order ◽  
Approximation Properties ◽  
Linear Combinations ◽  
Steklov Mean ◽  
Estimation Of Error ◽  
Direct Results

Abstract We study some direct results for the recently introduced family of modified Baskakov type operators. In particular, we obtain local direct results on ordinary and simultaneous approximation and an estimation of error for linear combinations in terms of higher order modulus of continuity. We have applied the Steklov mean as a tool for the linear approximating method.

scholarly journals Approximation properties of the new type generalized Bernstein-Kantorovich operators

2021 ◽  
Vol 7 (3) ◽  
pp. 3826-3844
Author(s):  
Mustafa Kara ◽  
Keyword(s):  
Rate Of Convergence ◽  
Modulus Of Continuity ◽  
Lipschitz Function ◽  
Type Theorem ◽  
Numerical Results ◽  
Bernstein Operators ◽  
Approximation Properties ◽  
Voronovskaya Type Theorem ◽  
New Type ◽  
Kantorovich Operators

<abstract><p>In this paper, we introduce new type of generalized Kantorovich variant of $ \alpha $-Bernstein operators and study their approximation properties. We obtain estimates of rate of convergence involving first and second order modulus of continuity and Lipschitz function are studied for these operators. Furthermore, we establish Voronovskaya type theorem of these operators. The last section is devoted to bivariate new type $ \alpha $-Bernstein-Kantorovich operators and their approximation behaviors. Also, some graphical illustrations and numerical results are provided.</p></abstract>

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