Quantitative Estimates of Voronovskaya Type in Weighted Spaces

Holhoş, Adrian

doi:10.1007/s00025-018-0814-9

Cited by 25 publications

(45 citation statements)

References 19 publications

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“…where is a positive constant and ∈ C 1 (see the study of Holhos 13 ) Following the previous study, 13 we can write, for any > 0, that…”

Section: Rate Of Convergencementioning

confidence: 97%

“…Now, defining the weight function ρ 1 in Theorem by ρ 1 ( x ) = 1 + x 2 on

double-struckR,

we consider the following weighted modulus of continuity:

w_{ρ_{1}} (f, δ) = \sup_{x, t \in R : |t - x| \leq δ} \frac{|f (t) - f (x)|}{ρ_{1} (t) + ρ_{1} (x)},

where δ is a positive constant and

f \in C_{ρ_{1}}

(see the study of Holhos)…”

Section: Rate Of Convergencementioning

confidence: 99%

“…Following the previous study, we can write, for any δ > 0, that

\begin{matrix} (||, f ((), t) - f false(x false)) & \leq ((), ρ_{1} ((), t) + ρ_{1} false(x false)) ((), 2 + ((), \frac{(||, t - x)}{δ})) w_{ρ_{1}} ((), f, δ) \\ \leq 4 ρ_{1} ((), t) ρ_{1} false(x false) ((), 1 + ((), \frac{φ_{x} ((), t)}{δ^{2}})) w_{ρ_{1}} ((), f, δ), \end{matrix}

where

φ_{x} ((), t) : = {((), t - x)}^{2} .

If we use the same operators L i in Theorem , we get, for all

f \in C_{ρ_{1}},

\begin{matrix} (||, T_{k} ((), f; x) - f false(x false)) & = (||, true \sum_{i = 1}^{k} a_{k - i + 1} L_{i} ((), f; x) - f false(x false)) \\ \leq true \sum_{i = 1}^{k} (||, a_{k - i + 1}) L_{i} ((), (||, f ((), t) - f false(x false)); x) + (||, f false(x false)) (||, true \sum_{i = 1}^{k} (||, a_{k - i + 1}) L_{i} ((), 1; x) - 1, <...>) \end{matrix}

…”

Section: Rate Of Convergencementioning

confidence: 99%

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Korovkin type approximation theorem via K_a−convergence on weighted spaces

Yıldız

Dirik

Demirci

2018

Math Methods in App Sciences

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“…where is a positive constant and ∈ C 1 (see the study of Holhos 13 ) Following the previous study, 13 we can write, for any > 0, that…”

Section: Rate Of Convergencementioning

confidence: 97%

“…Now, defining the weight function ρ 1 in Theorem by ρ 1 ( x ) = 1 + x 2 on

double-struckR,

we consider the following weighted modulus of continuity:

w_{ρ_{1}} (f, δ) = \sup_{x, t \in R : |t - x| \leq δ} \frac{|f (t) - f (x)|}{ρ_{1} (t) + ρ_{1} (x)},

where δ is a positive constant and

f \in C_{ρ_{1}}

(see the study of Holhos)…”

Section: Rate Of Convergencementioning

confidence: 99%

“…Following the previous study, we can write, for any δ > 0, that

\begin{matrix} (||, f ((), t) - f false(x false)) & \leq ((), ρ_{1} ((), t) + ρ_{1} false(x false)) ((), 2 + ((), \frac{(||, t - x)}{δ})) w_{ρ_{1}} ((), f, δ) \\ \leq 4 ρ_{1} ((), t) ρ_{1} false(x false) ((), 1 + ((), \frac{φ_{x} ((), t)}{δ^{2}})) w_{ρ_{1}} ((), f, δ), \end{matrix}

where

φ_{x} ((), t) : = {((), t - x)}^{2} .

If we use the same operators L i in Theorem , we get, for all

f \in C_{ρ_{1}},

\begin{matrix} (||, T_{k} ((), f; x) - f false(x false)) & = (||, true \sum_{i = 1}^{k} a_{k - i + 1} L_{i} ((), f; x) - f false(x false)) \\ \leq true \sum_{i = 1}^{k} (||, a_{k - i + 1}) L_{i} ((), (||, f ((), t) - f false(x false)); x) + (||, f false(x false)) (||, true \sum_{i = 1}^{k} (||, a_{k - i + 1}) L_{i} ((), 1; x) - 1, <...>) \end{matrix}

…”

Section: Rate Of Convergencementioning

confidence: 99%

See 1 more Smart Citation

Korovkin type approximation theorem via K_a−convergence on weighted spaces

Yıldız

Dirik

Demirci

2018

Math Methods in App Sciences

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“…Some estimates of the rate of convergence are available, by using a suitable modulus of continuity, introduced by Holhoş in [67]. More precisely, it is defined by setting, for every ∈ [0, ∞) and > 0,…”

Section: Journal Of Function Spacesmentioning

confidence: 99%

On Sequences of J. P. King-Type Operators

Acar

Montano

Garrancho

et al. 2019

Journal of Function Spaces

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This survey is devoted to a series of investigations developed in the last fifteen years, starting from the introduction of a sequence of positive linear operators which modify the classical Bernstein operators in order to reproduce constant functions and x2 on [0,1]. Nowadays, these operators are known as King operators, in honor of J. P. King who defined them, and they have been a source of inspiration for many scholars. In this paper we try to take stock of the situation and highlight the state of the art, hoping that this will be a useful tool for all people who intend to extend King’s approach to some new contents within Approximation Theory. In particular, we recall the main results concerning certain King-type modifications of two well known sequences of positive linear operators, the Bernstein operators and the Szász-Mirakyan operators.

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“…By using Grüss-type inequality for Bernstein operators, Gal and Gonska 11 proved a Voronovskaja-type theorem called as Grüss-Voronovskaja-type theorem. We can refer to other studies [12][13][14][15][16][17][18][19][20][21][22][23][24][25][26] and references therein for quantitative version of Voronovskaja-type theorem and Grüss-Voronovskaya-type theorem for some univariate and bivariate positive linear operators. Also, we remark that Voronovskaja-type theorem in asymptotic mean was studied by various authors (see, eg, Acar and Kajla 27 and Kajla et al 28 ).…”

Section: Introductionmentioning

confidence: 99%