Bernstein-type operators which preserve exactly two test functions

2013 ◽  
Vol 50 (4) ◽  
pp. 393-405 ◽  
Author(s):  
Ovidiu Pop ◽  
Dan Bǎrbosu ◽  
Petru Braica
Keyword(s):  
Convergence Theorem ◽  
General Class ◽  
Type Theorem ◽  
Positive Operators ◽  
Bernstein Operators ◽  
Test Functions ◽  
Approximation Properties ◽  
Bernstein Type ◽  
Voronovskaja Type Theorem

A general class of linear and positive operators dened by nite sum is constructed. Some of their approximation properties, including a convergence theorem and a Voronovskaja-type theorem are established. Next, the operators of the considered class which preserve exactly two test functions from the set {e0, e1, e2} are determined. It is proved that the test functions e0 and e1 are preserved only by the Bernstein operators, the test functions e0 and e2 only by the King operators while the test functions e1 and e2 only by the operators recently introduced by P. I. Braica, O. T. Pop and A. D. Indrea in [4].

scholarly journals About Some Linear Operators

2007 ◽  
Vol 2007 ◽  
pp. 1-13
Author(s):  
Ovidiu T. Pop
Keyword(s):  
Rate Of Convergence ◽  
General Class ◽  
Type Theorem ◽  
Modulus Of Smoothness ◽  
Positive Operators ◽  
Linear Operators ◽  
Linear Positive Operators ◽  
Voronovskaja Type Theorem

Using the method of Jakimovski and Leviatan from their work in 1969, we construct a general class of linear positive operators. We study the convergence, the evaluation for the rate of convergence in terms of the first modulus of smoothness and we give a Voronovskaja-type theorem for these operators.

scholarly journals The generalization of some results for Bernstein and Stancu operators

2011 ◽  
Vol 20 (2) ◽  
pp. 147-156
Author(s):  
DAN MICLAUS ◽  
◽  
PETRU I. BRAICA ◽  
Keyword(s):  
Uniform Convergence ◽  
Type Theorem ◽  
Bernstein Operators ◽  
Order Of Approximation ◽  
Test Functions ◽  
Stancu Operators ◽  
Voronovskaja Type Theorem

In the present paper we generalize some results for Bernstein and Stancu operators. Firstly, we establish a relationship between two results concerning calculation of test functions by Bernstein operators. Secondly, using this relationship and some known results we prove in every case a Voronovskaja type theorem, the uniform convergence and the order of approximation for Bernstein and Stancu operators.

scholarly journals Generalized p,q-Gamma-type operators

2020 ◽  
Vol 2020 ◽  
pp. 1-10
Author(s):  
Wen-Tao Cheng ◽  
Qing-Bo Cai
Keyword(s):  
Rate Of Convergence ◽  
Modulus Of Continuity ◽  
Type Theorem ◽  
Weighted Approximation ◽  
Local Approximation ◽  
Pointwise Estimates ◽  
Approximation Properties ◽  
Central Moments ◽  
Voronovskaja Type Theorem

In the present paper, the generalized p,q-gamma-type operators based on p,q-calculus are introduced. The moments and central moments are obtained, and some local approximation properties of these operators are investigated by means of modulus of continuity and Peetre K-functional. Also, the rate of convergence, weighted approximation, and pointwise estimates of these operators are studied. Finally, a Voronovskaja-type theorem is presented.

scholarly journals On the Approximation Properties of q − Analogue Bivariate λ -Bernstein Type Operators

2020 ◽  
Vol 2020 ◽  
pp. 1-11
Author(s):  
Edmond Aliaga ◽  
Behar Baxhaku
Keyword(s):  
Type Theorem ◽  
Degree Of Approximation ◽  
Modulus Of Smoothness ◽  
Approximation Properties ◽  
Bernstein Type ◽  
Matlab Software ◽  
Type Space ◽  
Mixed Modulus Of Smoothness ◽  
Mixed Modulus ◽  
Boolean Sum

In this article, we establish an extension of the bivariate generalization of the q -Bernstein type operators involving parameter λ and extension of GBS (Generalized Boolean Sum) operators of bivariate q -Bernstein type. For the first operators, we state the Volkov-type theorem and we obtain a Voronovskaja type and investigate the degree of approximation by means of the Lipschitz type space. For the GBS type operators, we establish their degree of approximation in terms of the mixed modulus of smoothness. The comparison of convergence of the bivariate q -Bernstein type operators based on parameters and its GBS type operators is shown by illustrative graphics using MATLAB software.

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>

scholarly journals Approximation properties of (p, q)-Bernstein type operators

2016 ◽  
Vol 8 (2) ◽  
pp. 222-232 ◽  
Author(s):  
Zoltán Finta
Keyword(s):  
Rate Of Convergence ◽  
Quantitative Estimation ◽  
Bernstein Operators ◽  
Bernstein Operator ◽  
Approximation Properties ◽  
Bernstein Type ◽  
Direct Approximation

AbstractWe introduce a new generalization of the q-Bernstein operators involving (p, q)-integers, and we establish some direct approximation results. Further, we define the limit (p, q)-Bernstein operator, and we obtain its estimation for the rate of convergence. Finally, we introduce the (p, q)-Kantorovich type operators, and we give a quantitative estimation.

scholarly journals Approximation properties of modified Srivastava-Gupta operators based on certain parameter

2018 ◽  
Vol 38 (1) ◽  
pp. 41-53 ◽  
Author(s):  
Alok Kumar ◽  
Dr Vandana
Keyword(s):  
Maximal Function ◽  
Statistical Convergence ◽  
Approximation Theorem ◽  
Type Theorem ◽  
Weighted Approximation ◽  
Linear Functions ◽  
Approximation Properties ◽  
Voronovskaja Type Theorem ◽  
Direct Approximation ◽  
Lipschitz Type Maximal Function

In the present article, we give a modified form of generalized Srivastava-Gupta operators based on certain parameter which preserve the constant as well as linear functions. First, we estimate moments of the operators and then prove Voronovskaja type theorem. Next, direct approximation theorem, rate of convergence and weighted approximation by these operators in terms of modulus of continuity are studied. Then, we obtain point-wise estimate using the Lipschitz type maximal function. Finaly, we study the $A$-statistical convergence of these operators.

scholarly journals On Stancu-Type Generalization of Modified p , q -Szász-Mirakjan-Kantorovich Operators

2021 ◽  
Vol 2021 ◽  
pp. 1-11
Author(s):  
Yong-Mo Hu ◽  
Wen-Tao Cheng ◽  
Chun-Yan Gui ◽  
Wen-Hui Zhang
Keyword(s):  
Rate Of Convergence ◽  
Present Article ◽  
Modulus Of Continuity ◽  
Type Theorem ◽  
Weighted Approximation ◽  
Local Approximation ◽  
Pointwise Estimates ◽  
Approximation Properties ◽  
Voronovskaja Type Theorem ◽  
Kantorovich Operators

In the present article, we construct p , q -Szász-Mirakjan-Kantorovich-Stancu operators with three parameters λ , α , β . First, the moments and central moments are estimated. Then, local approximation properties of these operators are established via K -functionals and Steklov mean in means of modulus of continuity. Also, a Voronovskaja-type theorem is presented. Finally, the pointwise estimates, rate of convergence, and weighted approximation of these operators are studied.

scholarly journals On Some Approximation Properties for a Sequence of λ-Bernstein Type Operators

2021 ◽  
pp. 4903-4915
Author(s):  
Ali Jassim Muhammad ◽  
Asma Jaber
Keyword(s):  
Asymptotic Formula ◽  
Uniform Convergence ◽  
Convergence Theorem ◽  
Recurrence Relations ◽  
Bernstein Polynomials ◽  
Negative Integer ◽  
Asymptotic Formulas ◽  
Order Moment ◽  
Approximation Properties ◽  
Bernstein Type

In 2010, Long and Zeng introduced a new generalization of the Bernstein polynomials that depends on a parameter  and called -Bernstein polynomials. After that, in 2018, Lain and Zhou studied the uniform convergence for these -polynomials and obtained a Voronovaskaja-type asymptotic formula in ordinary approximation. This paper studies the convergence theorem and gives two Voronovaskaja-type asymptotic formulas of the sequence of -Bernstein polynomials in both ordinary and simultaneous approximations. For this purpose, we discuss the possibility of finding the recurrence relations of the -th order moment for these polynomials and evaluate the values of -Bernstein for the functions ,  is a non-negative integer

scholarly journals Note on a Schurer-Stancu-type operator

2015 ◽  
Vol 24 (1) ◽  
pp. 61-67
Author(s):  
ADRIAN D. INDREA ◽  
◽  
ANAMARIA INDREA ◽  
PETRU I. BRAICA ◽  
◽  
...  
Keyword(s):  
Type Theorem ◽  
Type Operator ◽  
Test Functions ◽  
New Class ◽  
Voronovskaja Type Theorem

The aim of this paper is to introduce a class of operators of Schurer-Stancu-type with the property that the test functions e0 and e1 are reproduced. Also, in our approach, a theorem of error approximation and a Voronovskaja-type theorem for this operators are obtained. Finally, we study the convergence of the iterates for our new class of operators.

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