Approximation by Gamma type operators

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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“…For papers inspired by [38,39] we refer the readers to [40][41][42] and [43][44][45][46], respectively.…”

Section: A Brief Historymentioning

confidence: 99%

On Sequences of J. P. King-Type Operators

Acar

Montano

Garrancho

et al. 2019

Journal of Function Spaces

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“…In past years, there have been several modifications of operators to enhance their convergence and error estimation process (see previous studies 6‐8 ). In 2003, King 9 presented a sequence of linear positive operators which approximated each continuous function on [0, 1] while preserving the test function x 2 .…”

Section: Introductionmentioning

confidence: 99%

On the preservation of functions with exponential growth by modified Ismail–May operators

Mishra

Deo

2021

Math Methods in App Sciences

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In this study, our aim is to provide a modification of the so‐called Ismail–May operators that preserve exponential functions eAx,.7emA.5em∈.5emℝ. In support of this, we begin by estimating the convergence rate of the operators in terms of usual and exponential modulus of continuity. We also provide a global approximation and a quantitative Voronovskaya result. Moreover, to validate the modification, we exhibit some graphical representations using Mathematica software to compare the original operator and its modification. We conclude that the modified operators not only preserve exponential functions but also provide faster rate of convergence for A > 0.

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“…In [11], Zhou et al extended the works of Cheng et al in [10] and constructed ðp, qÞ-Gamma-Stancu operators. There are many papers about the research and application of other Gamma-type operators, and we mention some of them [12][13][14][15][16][17].…”

Section: Introductionmentioning

confidence: 99%

Approximation Properties of New Modified Gamma Operators

Cheng

Zhou

et al. 2021

Journal of Function Spaces

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This paper is aimed at constructing new modified Gamma operators using the second central moment of the classic Gamma operators. And we will compute the first, second, fourth, and sixth order central moments by the moment computation formulas, and their quantitative properties are researched. Then, the global results are established in certain weighted spaces and the direct results including the Voronovskaya-type asymptotic formula, and point-wise estimates are investigated. Also, weighted approximation of these operators is discussed. Finally, the quantitative Voronovskaya-type asymptotic formula and Grüss Voronovskaya-type approximation are presented.

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Approximation by Gamma type operators

Cited by 16 publications

References 14 publications

On Sequences of J. P. King-Type Operators

On Sequences of J. P. King-Type Operators

On the preservation of functions with exponential growth by modified Ismail–May operators

Approximation Properties of New Modified Gamma Operators

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