New estimates for the differences of positive linear operators

The main object of this paper is to improve some of the known estimates for classical Kantorovich operators. A quantitative Voronovskaya-type result in terms of second moduli of continuity which improves some previous results is obtained. In order to explain nonmultiplicativity of the Kantorovich operators a Chebyshev-Grüss inequality is given. Two Grüss-Voronovskaya theorems for Kantorovich operators are considered as well.

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“…Using [7, Lemma 3.1] for r = 1, s = 2, f h,3 = u and g h, 3 = v, for all h ∈ (0, 1] and n ∈ N, it follows…”

mentioning

confidence: 99%

Classical Kantorovich Operators Revisited

Acu

Gonska

2019

Ukr Math J

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“…The classical Durrmeyer operators M n are obtained for α = β = 0. In order to give the estimate for the difference of the Durrmeyer operators we need the following result (see, e.g., [2]): Then, for each ϕ ∈ C 2 [0, 1] there is ξ ∈ [0, 1] such that…”

Section: The Durrmeyer Operators With Jacobi Weightsmentioning

confidence: 99%

Estimates for the differences of positive linear operators and their derivatives

Acu

Raşa

2019

Numer Algor

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The present paper deals with the estimate of the differences of certain positive linear operators and their derivatives. Our approach involves operators defined on bounded intervals, as Bernstein operators, Kantorovich operators, genuine Bernstein-Durrmeyer operators, Durrmeyer operators with Jacobi weights. The estimates in quantitative form are given in terms of first modulus of continuity. In order to analyze the theoretical results in the last section we consider some numerical examples.

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“…Gonska et al and Gonska and Raşa established the estimates on the differences of positive linear operators. Acu and Raşa developed an approach suitable for operators constructed with the same fundamental functions, and Aral et al discussed about the difference of the operators in weighted space. Also, some of the results on this topic are collected in the recent book by Gupta et al In terms of modulus of continuity of f and

f^{''}

, some results have been estimated .…”

Section: Differences Of Operatorsmentioning

confidence: 99%

“…Denote

b^{F} : = F (e_{1}), μ_{r}^{F} = F {(e_{1} - b^{F} e_{0})}^{r}, r \in N .

Let

ω_{1} (f, h) : = \sup \{|f (x + h) - f (x)| : | δ | \leq h, x, x + h \in I\}

be the classical modulus of smoothness given for a compact interval I and

h \in {double-struckR}_{+}

. For the functional F , the following basic results were obtained …”

Section: Differences Of Operatorsmentioning

confidence: 99%

“…Remark We may remark here that in theorem 3 in the work of Acu and Raşa, the authors considered

δ false(x false) = \sup_{k \in K} (||, b^{F_{k}} - b^{G_{k}})

, which is true in general only when summation index K is finite, ie,

\sum_{k \in K} = \sum_{k = 0}^{n}

, but in case

\sum_{k \in K} = \sum_{k = 0}^{\infty}

, the value of δ ( x ) becomes infinite. Second, for λ > 0, we know ω ( f , λδ ) ≤ ( λ + 1) ω ( f , δ ).…”

Section: Differences Of Operatorsmentioning

confidence: 99%

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General estimates for the difference of operators

Gupta

2019

Comp and Math Methods

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This paper deals with the quantitative estimates for the difference of operators having the same basis functions. The results essentially improve the earlier known results. In the end, some applications for the differences of Szász-Mirakyan operators with its different variants are presented. KEYWORDS difference of operators, modulus of continuity, positive linear operators, Szász-Mirakyan operators 1 DIFFERENCES OF OPERATORS Gonska et al 1,2 and Gonska and Raşa 3 established the estimates on the differences of positive linear operators. Acu and Raşa 4 developed an approach suitable for operators constructed with the same fundamental functions, and Aral et al 5 discussed about the difference of the operators in weighted space. Also, some of the results on this topic are collected in the recent book by Gupta et al. 6

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New estimates for the differences of positive linear operators

Cited by 55 publications

References 4 publications

Classical Kantorovich Operators Revisited

Classical Kantorovich Operators Revisited

Estimates for the differences of positive linear operators and their derivatives

General estimates for the difference of operators

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