A Bernstein-type operator approximating continuous functions on the semi-axis

Bleimann, G.; Butzer, Paul L.; Hahn, L.

doi:10.1016/1385-7258(80)90027-x

Cited by 117 publications

(49 citation statements)

References 6 publications

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“…Now it is easy to see that from (2.9), 8) and since [n + 1]→∞, q n →1 as n→∞, condition (3.7) holds for υ = 1.…”

Section: Properties Of the Operatorsmentioning

confidence: 99%

“…In [8], Bleimann, Butzer, and Hahn introduced the following operators: 1 + t)) ν for ν = 0,1,2. Some generalization of the operators (1.1) were given in [11][12][13].…”

Section: Introductionmentioning

confidence: 99%

“…In [4], Barbasu gave Stancu-type generalization of these operators and II'inskii and Ostrovska [5] studied their different convergence properties. Also some results on the statistical and ordinary approximation of functions by Meyer-König and Zeller operators based on q-integers can be found in [6,7], respectively.In [8], Bleimann, Butzer, and Hahn introduced the following operators: 1 + t)) ν for ν = 0,1,2. Some generalization of the operators (1.1) were given in [11][12][13].…”

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confidence: 99%

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Bleimann, Butzer, and Hahn Operators Based on the -Integers

Aral

Doğru

2007

J Inequal Appl

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We give a new generalization of Bleimann, Butzer, and Hahn operators, which includes qintegers. We investigate uniform approximation of these new operators on some subspace of bounded and continuous functions. In Section 3, we show that the rates of convergence of the new operators in uniform norm are better than the classical ones. We also obtain a pointwise estimation in a general Lipschitz-type maximal function space. Finally, we de?fine a generalization of these new operators and study the uniform convergence of them.Copyright © 2007 A. Aral and O. Dogru. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited. IntroductionRecently, in 1997, Phillips [1] used the q-integers in approximation theory where it is considered q-based generalization of classical Bernstein polynomials. It was obtained by replacing the binomial expansion with the general one, the q-binomial expansion. Phillips has obtained the rate of convergence and Voronovskaja-type asymptotic formulae for these new Bernstein operators based on q-integers. Later, some results are established in due course by Phillips et al. (see [2,3,1]). In [4], Barbasu gave Stancu-type generalization of these operators and II'inskii and Ostrovska [5] studied their different convergence properties. Also some results on the statistical and ordinary approximation of functions by Meyer-König and Zeller operators based on q-integers can be found in [6,7], respectively.In [8], Bleimann, Butzer, and Hahn introduced the following operators: 1 + t)) ν for ν = 0,1,2. Some generalization of the operators (1.1) were given in [11][12][13].In this paper, we derive a q-integers-type modification of BBH operators that we call q-BBH operators and investigate their Korovkin-type approximation properties by using the test function (t/(1 + t)) ν for ν = 0,1,2. Also, we define a space of generalized Lipschitz-

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“…Now it is easy to see that from (2.9), 8) and since [n + 1]→∞, q n →1 as n→∞, condition (3.7) holds for υ = 1.…”

Section: Properties Of the Operatorsmentioning

confidence: 99%

“…In [8], Bleimann, Butzer, and Hahn introduced the following operators: 1 + t)) ν for ν = 0,1,2. Some generalization of the operators (1.1) were given in [11][12][13].…”

Section: Introductionmentioning

confidence: 99%

mentioning

confidence: 99%

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Bleimann, Butzer, and Hahn Operators Based on the -Integers

Aral

Doğru

2007

J Inequal Appl

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“…It should be remarked that letting r = 1, b n = 1 and a 1,n = 1 for all n ∈ N, in (2.1) we get the well known BBH operators (see [2]) defined on the space…”

Section: Construction Of the Operatorsmentioning

confidence: 99%

Korovkin type approximation theorem for BBH type operators via J-convergence

Aktuğlu

Özarslan

2010

Mathematica Slovaca

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“…−n x k for k = 0,1,...,n, u n,k = 0 for k > n, and we choose the net Δ n = (k/(n − k + 1)) k=0,n , then L n , n ∈ N, become the well-known Bleimann-Butzer-Hahn operators [3].…”

Section: Construction Of the Class (Lmentioning

confidence: 99%

Linear operators that preserve some test functions

Agratini

2006

International Journal of Mathematics and Mathematical Sciences

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The paper centers around a pair of sequences of linear positive operators. The former has the degree of exactness one and the latter has its degree of exactness null, but, as a novelty, it reproduces the third test function of Korovkin theorem. Quantitative estimates of the rate of convergence are given in different function spaces traveling from classical approximation to approximation in abstract spaces. Particular classes are also studied.

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A Bernstein-type operator approximating continuous functions on the semi-axis

Cited by 117 publications

References 6 publications

Bleimann, Butzer, and Hahn Operators Based on the -Integers

Bleimann, Butzer, and Hahn Operators Based on the -Integers

Korovkin type approximation theorem for BBH type operators via J-convergence

Linear operators that preserve some test functions

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