scholarly journals On the Durrmeyer-Type Variant and Generalizations of Lototsky–Bernstein Operators

Symmetry ◽  
2021 ◽  
Vol 13 (10) ◽  
pp. 1841
Author(s):  
Ulrich Abel ◽  
Octavian Agratini
Keyword(s):  
Convergence Speed ◽  
Positive Operators ◽  
Moduli Of Smoothness ◽  
Bernstein Operators ◽  
Approximation Properties ◽  
New Class

The starting points of the paper are the classic Lototsky–Bernstein operators. We present an integral Durrmeyer-type extension and investigate some approximation properties of this new class. The evaluation of the convergence speed is performed both with moduli of smoothness and with K-functionals of the Peetre-type. In a distinct section we indicate a generalization of these operators that is useful in approximating vector functions with real values defined on the hypercube [0,1]q, q>1. The study involves achieving a parallelism between different classes of linear and positive operators, which will highlight a symmetry between these approximation processes.

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 Positive operators and approximation in function spaces on completely regular spaces

2003 ◽  
Vol 2003 (61) ◽  
pp. 3841-3871 ◽  
Author(s):  
Francesco Altomare ◽  
Sabrina Diomede
Keyword(s):  
Natural Extension ◽  
Function Spaces ◽  
Positive Operators ◽  
Linear Operators ◽  
Topological Spaces ◽  
Approximation Properties ◽  
Completely Regular ◽  
Borel Measures ◽  
Korovkin Type Theorems ◽  
Locally Convex

We discuss the approximation properties of nets of positive linear operators acting on function spaces defined on Hausdorff completely regular spaces. A particular attention is devoted to positive operators which are defined in terms of integrals with respect to a given family of Borel measures. We present several applications which, in particular, show the advantages of such a general approach. Among other things, some new Korovkin-type theorems on function spaces on arbitrary topological spaces are obtained. Finally, a natural extension of the so-called Bernstein-Schnabl operators for convex (not necessarily compact) subsets of a locally convex space is presented as well.

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 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>

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.

Approximation by Stancu–Chlodowsky type λ-Bernstein operators

2020 ◽  
Vol 26 (1) ◽  
pp. 97-110
Author(s):  
M. Mursaleen ◽  
A. A. H. Al-Abied ◽  
M. A. Salman
Keyword(s):  
Asymptotic Formula ◽  
Weighted Space ◽  
Direct Result ◽  
Bernstein Operators ◽  
Approximation Properties ◽  
Convergence Properties ◽  
Korovkin’S Theorem

AbstractIn this paper, we give some approximation properties by Stancu–Chlodowsky type λ-Bernstein operators in the polynomial weighted space and obtain the convergence properties of these operators by using Korovkin’s theorem. We also establish the direct result and the Voronovskaja type asymptotic formula.

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 Certain Summation Integral Type Operators

2015 ◽  
Vol 48 (1) ◽  
Author(s):  
P. Patel ◽  
Vishnu Narayan Mishra
Keyword(s):  
Asymptotic Formula ◽  
Rate Of Convergence ◽  
Approximation Theorem ◽  
Weighted Approximation ◽  
Positive Operators ◽  
Inverse Theorem ◽  
Approximation Properties ◽  
Integral Type ◽  
Linear Positive Operators ◽  
Direct Results

AbstractIn the present paper, we study approximation properties of a family of linear positive operators and establish direct results, asymptotic formula, rate of convergence, weighted approximation theorem, inverse theorem and better approximation for this family of linear positive operators.

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