Note on a Class of Operators on Infinite Interval

Agratini, Octavian

doi:10.1515/dema-1999-0414

Cited by 2 publications

(2 citation statements)

References 4 publications

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“…Now we will give an estimate concerning the rate of convergence as given in [8,52,109]. We define the space of general Lipschitz-type maximal functions on…”

Section: Then For Any Function F In H ω One Hasmentioning

confidence: 99%

Applications of q-Calculus in Operator Theory

Aral¹,

Gupta²,

Agarwal³

2013

372

206

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In order to approximate integrable functions on the interval [0, 1], Kantorovich gave modified Bernstein polynomials. Later in the year 1967 Durrmeyer [58] considered a more general integral modification of the classical Bernstein polynomials, which were studied first by Derriennic [47]. Also some other generalizations of the Bernstein polynomials are available in the literature. The other most popular generalization as considered by Goodman and Sharma [82], namely, genuine Bernstein-Durrmeyer operators. In this chapter we discuss the q analogues of various integral modifications of Bernstein polynomials. The results were discussed in recent papers [45, 62, 86, 89, 92, 94, 121], etc. q-Bernstein-Kantorovich OperatorsRecently, Dalmanoglu [45] proposed the q-Kantorovich-Bernstein operators asIn case q = 1, the operators (4.1) reduce to well-known Bernstein-Kantorovich operators Direct ResultsFor the operators (4.1), Dalmanoglu [45] obtained the following theorems:Theorem 4.1. If the sequence (q n ) satisfies the conditions lim n→∞ q n = 1 and lim n→∞Proof. First, we haveAlso by definition of q-integral q-Bernstein-Kantorovich Operators 115Again by definition of q-integralTo estimate K n,q (t 2 , x), we haveTherefore using [k + 1] q = q[k] q + 1 and using the similar methods as above, we haveReplacing q by a sequence {q n } such that lim n→∞ q n = 1, it is easily seen that K n,q (t i , x), i = 0, 1, 2 converges uniformly to t i . Thus the result follows by Korovkin's theorem. for all f ∈ C[0, a] and δ n = K n,q ((t − x) 2 , x).

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“…Now we will give an estimate concerning the rate of convergence as given in [8,52,109]. We define the space of general Lipschitz-type maximal functions on…”

Section: Then For Any Function F In H ω One Hasmentioning

confidence: 99%

Applications of q-Calculus in Operator Theory

Aral¹,

Gupta²,

Agarwal³

2013

372

206

View full text Add to dashboard Cite

show abstract

“…Aral and O. Dogru 7 Now we will give an estimate concerning the rate of convergence as given in [13,15,16]. We define the space of general Lipschitz-type maximal functions on…”

Section: Properties Of the Operatorsmentioning

confidence: 99%

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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Note on a Class of Operators on Infinite Interval

Cited by 2 publications

References 4 publications

Applications of q-Calculus in Operator Theory

Applications of q-Calculus in Operator Theory

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

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