Generalization of Szasz operators involving Brenke type polynomials

Varma, Serhan; Sucu, Sezgin; İçöz, Gürhan

doi:10.1016/j.camwa.2012.01.025

Cited by 68 publications

(47 citation statements)

References 5 publications

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“…For B(v) = e v , A(v) = 1 and w(v) = 1, using relation (6), these operators include the Szász-Baskakov type operators given by (4). The aim of the paper is to investigate uniform convergence theorem, error estimates by means of moduli of continuity, Voronovskaya-type asymptotic theorem, quantitative-Voronovskaya result, Grüss Voronovskaya-type theorem and an error estimate for functions with derivatives of bounded variation.…”

Section: Introductionmentioning

confidence: 99%

Baskakov‐Durrmeyer type operators involving generalized Appell Polynomials

Neer

Acu

Agrawal

2019

Math Methods in App Sciences

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In this paper, we define the Baskakov‐Durrmeyer type operators based on generalized Appell polynomials. Here, we establish moment estimates, an estimate via weighted modulus of continuity and a Voronovoskaya type asymptotic result. Further, we study a quantitative‐Voronovoskaya‐type theorem and Grüss Voronovskaya‐type theorem. Lastly, we give the approximation result for functions having derivatives of bounded variation.

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Section: Introductionmentioning

confidence: 99%

Baskakov‐Durrmeyer type operators involving generalized Appell Polynomials

Neer

Acu

Agrawal

2019

Math Methods in App Sciences

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“…In 1969, Jakimovski and Leviatan gave a generalization of Szász operators by using the Appell polynomials

h_{j} false(x false) = true \sum_{i = 0}^{j} a_{i} \frac{x^{j - i}}{false(j - i false)!}

which satisfy the identity

g false(t false) e^{t x} = true \sum_{j = 0}^{\infty} h_{j} false(x false) t^{j} .

Here,

g false(z false) = \sum_{k = 0}^{\infty} a_{k} z^{k}

is an analytic function in the disc | z | < R ,( R > 1) and g (1) ≠ 0. Varma et al constructed positive linear operators based on orthogonal polynomials, eg, Brenke polynomials. Suppose that

scriptG false(t false) = true \sum_{j = 0}^{\infty} a_{k} z^{j}, scriptG false(0 false) \neq 0, scriptH false(t false) = true \sum_{j = 0}^{\infty} b_{j} z^{j}, scriptH false(0 false) \neq 0

be analytic functions in the disc | z | < R ,( R > 1) where a j and b j are real.…”

Section: Introductionmentioning

confidence: 99%

“…Suppose that

scriptG false(t false) = true \sum_{j = 0}^{\infty} a_{k} z^{j}, scriptG false(0 false) \neq 0, scriptH false(t false) = true \sum_{j = 0}^{\infty} b_{j} z^{j}, scriptH false(0 false) \neq 0

be analytic functions in the disc | z | < R ,( R > 1) where a j and b j are real. The generating function for these polynomials is given by

scriptG false(t false) scriptH false(t x false) = true \sum_{j = 0}^{\infty} h_{j} false(x false) t^{j}

from which the explicit form of h j ( x ) is the following:

h_{j} false(x false) = true \sum_{i = 0}^{j} a_{j - i} b_{j} x^{j}, j = 0, 1, 2, ⋯ .

Varma et al proposed Szász operators involving the Brenke polynomials defined by

A_{η} false(normalΦ; x false) = \frac{1}{scriptG false(1 false) scriptH false(η x false)} true \sum_{k = 0}^{\infty} h_{k} false(η x false) normalΦ ((), \frac{k}{η}),

where x ≥ 0 and

η \in double-struckN

. Atakut and Büyükyzici discussed the convergence and approximation properties of the Kantorovich‐Szász–type operators defined as

{scriptK}_{η}^{α_{η}, β_{η}} false(normalΦ; x false) = \frac{β_{η}}{scriptG false(1 false) scriptH false(α_{η} x false)} true \sum<...>

…”

Section: Introductionmentioning

confidence: 99%

Approximation by a sequence of operators of the Srivastava‐Gupta type involving the Brenke polynomials with a certain parameter

Baxhaku

Berisha

2019

Math Methods in App Sciences

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We introduce a new sequence of linear positive operators by combining the Brenke polynomials and the Srivastava‐Gupta–type operators defined by Srivastava‐Gupta. obtain the moments of the operators and present some classical and statistical approximation properties by means of Korovkin results. Next, we estimate a global result, which includes the Voronovskaya‐type asymptotic formula, local approximation, error estimation in terms of weighted modulus of continuity, and for functions in a Lipschitz‐type space. Lastly, we estimate the rate of approximation for functions with derivatives of bounded variation.

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“…This operator is a generalization of Bernstein polynomials to the infinite interval. Szász operators and their generalizations have been studied by many authors (see [1,[4][5][6][7][8][9][10][12][13][14][15]). One of the generalizations of the Szász operator including parameters a n and b n was given byİspir and Atakut [7] as follows:…”

Section: Introductionmentioning

confidence: 99%