Szász–Mirakyan Type Operators Which Fix Exponentials

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In this study, we introduce newly defined Gamma operators which preserve constants and e2μ·, μ>0 functions. In accordance with this purpose, we focus on their approximation properties such as uniform convergence, rate of convergence, asymptotic formula, and saturation results. Superior properties of introduced operators have been tested both theoretically and numerically in certain senses to highlight the performance of the new constructions of Gamma operators.

Section: Introductionmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

Approximation by Gamma type operators

Deveci

Acar

Alagöz

2020

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“…In recent years, various operators were conveniently modified, which preserve the test functions emerged in this field and a lot of advancements have been made regarding this subject to acquire better approximation. For some recent studies on linear positive operators preserving exponential functions, we refer the readers to the works of Acar et al, Yilmaz et al, Gupta and Aral, Bessenyei and Páles, and Murat et al…”

Section: Introductionmentioning

confidence: 99%

Reconstruction of Baskakov operators preserving some exponential functions

Ozsarac

Acar

2018

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The present paper deals with a new modification of Baskakov operators in which the functions exp(μt) and exp(2μt), μ>0 are preserved. Approximation properties of the operators are captured, ie, uniform convergence and rate of convergence of the operators in terms of modulus of continuity, approximation behaviors of the operators exponential weighted spaces, and pointwise convergence of the operators by means of the Voronovskaya theorem. Advantages of the operators for some special functions are presented.

“…Since these operators are not suitable for approximating discontinuous functions to obtain an approximation process in spaces of integrable functions on unbounded intervals, Butzer introduced and studied an integral modification of the operators S n , the so‐called Szász‐Mirakyan‐Kantorovich operators defined by

K_{n} (f, x) = n e^{- n x} \sum_{k = 0}^{\infty} \frac{{(n x)}^{k}}{k!} \int_{\frac{k}{n}}^{\frac{k + 1}{n}} f (t) d t, x \geq 0, n \in N .

Many researchers have developed relevant studies on these operators, and numerous articles can be mentioned, interrelated with Kantorovich research: see, eg, Ditzian and Totik and Duman et al In the literature, there are a lot of studies that involve Szàsz operators, Szàsz‐Kantorovich operators, and their generalizations. Some of them are Duman et al, Aral et al, Acar et al, Boyanov and Veselinov, and Gupta and Aral . We are now concerned only with the known results which are necessary for this study.…”

Section: Introductionmentioning

confidence: 99%

“…Recently, Aral et al constructed Szàsz‐Mirakyan type operators that reproduce the exponential functions

\exp false(μ t false)

and

\exp false(2 μ t false)

, μ > 0 and shown that the operators perform better than S n under sufficient conditions. These generalized Szàsz operators are defined by

G_{n} (f, x) = e^{- n α_{n} (x)} \sum_{k = 0}^{\infty} \frac{{(n α_{n} (x))}^{k}}{k!} e^{μ x} e^{- μ \frac{k}{n}} f (\frac{k}{n}), x \geq 0, n \in N

where

α_{n} ((), x) = \frac{μ x}{n ((), e^{\frac{μ}{n}} - 1)},

for every

f \in C ([), 0, \infty)

for which the series at the right‐hand side is absolutely convergent.…”

Section: Introductionmentioning

confidence: 99%

Approximation properties of Szász‐Mirakyan‐Kantorovich type operators

Aral

Limmam

Ozsarac

2018

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In this paper, we introduce and study new type Szász‐Mirakyan‐Kantorovich operators using a technique different from classical one. This allow to analyze the mentioned operators in terms of exponential test functions instead of the usual polynomial type functions. As a first result, we prove Korovkin type approximation theorems through exponential weighted convergence. The rate of convergence of the operators is obtained for exponential weights.