Bernstein-type operators which preserve polynomials

Cárdenas-Morales, Daniel; Garrancho, Pedro; Raşa, Ioan

doi:10.1016/j.camwa.2011.04.063

Cited by 90 publications

(63 citation statements)

References 8 publications

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“…Proof Using Lemma and the fact that { e 0 , τ , τ 2 } is an extended complete Tchebychev system on [0,1] (see Cárdenas‐Morales et al), it follows that

{scriptU}_{n}^{ρ, τ}

converges uniformly to g on [0,1]. □…”

Section: Auxiliary Resultsmentioning

confidence: 94%

“…For → ∞ the sequence  , n converges uniformly to the modified Bernstein polynomials B n studied by Cárdenas-Morales et al 10 Corollary 5.1. Let g ∈ C 2 [0, 1] and x ∈ [0, 1].…”

Section: The Modified Bernstein Operatorsmentioning

confidence: 87%

“…For ρ → ∞ the sequence

{scriptU}_{n}^{ρ, τ}

converges uniformly to the modified Bernstein polynomials

B_{n}^{τ}

studied by Cárdenas‐Morales et al…”

Section: Applicationsmentioning

confidence: 88%

“…Let τ be a continuously infinite times differentiable function verifying the condition τ (1) = 1, τ (0) = 0 and τ ′ ( x ) > 0 for x ∈ [0,1]. Recently, Cárdenas‐Morales et al considered a new Bernstein‐type operator defined for g ∈ C [0,1] by

B_{n}^{τ} g : = B_{n} false(g \circ τ^{- 1} false) \circ τ

, namely,

B_{n}^{τ} (g; x) = \sum_{k = 0}^{n} (g \circ τ^{- 1}) (\frac{k}{n}) p_{n, k}^{τ} (x),

where

p_{n, k}^{τ} false(x false) = ((), \begin{matrix} center \\ array n \\ array k \end{matrix}) {false(τ false(x false) false)}^{k} {false(1 - τ false(x false) false)}^{n - k}, x \in false[0, 1 false] .

The modified Bernstein operators

B_{n}^{τ}

have a better degree of approximation depending on τ .…”

Section: Introductionmentioning

confidence: 99%

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Certain positive linear operators with better approximation properties

Raţiu

Acu

Acar

et al. 2018

Math Methods in App Sciences

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The present paper deals with a new positive linear operator which gives a connection between the Bernstein operators and their genuine Bernstein‐Durrmeyer variants. These new operators depend on a certain function τ defined on [0,1] and improve the classical results in some particular cases. Some approximation properties of the new operators in terms of first and second modulus of continuity and the Ditzian‐Totik modulus of smoothness are studied. Quantitative Voronovskaja–type theorems and Grüss‐Voronovskaja–type theorems constitute a great deal of interest of the present work. Some numerical results that compare the rate of convergence of these operators with the classical ones and illustrate the relevance of the theoretical results are given.

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“…Proof Using Lemma and the fact that { e 0 , τ , τ 2 } is an extended complete Tchebychev system on [0,1] (see Cárdenas‐Morales et al), it follows that

{scriptU}_{n}^{ρ, τ}

converges uniformly to g on [0,1]. □…”

Section: Auxiliary Resultsmentioning

confidence: 94%

“…For → ∞ the sequence  , n converges uniformly to the modified Bernstein polynomials B n studied by Cárdenas-Morales et al 10 Corollary 5.1. Let g ∈ C 2 [0, 1] and x ∈ [0, 1].…”

Section: The Modified Bernstein Operatorsmentioning

confidence: 87%

“…For ρ → ∞ the sequence

{scriptU}_{n}^{ρ, τ}

converges uniformly to the modified Bernstein polynomials

B_{n}^{τ}

studied by Cárdenas‐Morales et al…”

Section: Applicationsmentioning

confidence: 88%

B_{n}^{τ} g : = B_{n} false(g \circ τ^{- 1} false) \circ τ

, namely,

B_{n}^{τ} (g; x) = \sum_{k = 0}^{n} (g \circ τ^{- 1}) (\frac{k}{n}) p_{n, k}^{τ} (x),

where

p_{n, k}^{τ} false(x false) = ((), \begin{matrix} center \\ array n \\ array k \end{matrix}) {false(τ false(x false) false)}^{k} {false(1 - τ false(x false) false)}^{n - k}, x \in false[0, 1 false] .

The modified Bernstein operators

B_{n}^{τ}

have a better degree of approximation depending on τ .…”

Section: Introductionmentioning

confidence: 99%

See 2 more Smart Citations

Certain positive linear operators with better approximation properties

Raţiu

Acu

Acar

et al. 2018

Math Methods in App Sciences

View full text Add to dashboard Cite

show abstract

“…The Markov operators which preserve the monomial e 2 are named King type operators. They have been approached in recent years, starting with the paper of King [14] (see also [1,10,3,8,4]). …”

Section: Introductionmentioning

confidence: 99%