Approximation on parametric extension of Baskakov–Durrmeyer operators on weighted spaces

Nasiruzzaman, Md.; Rao, Nadeem; Wazir, Samar; Kumar, Ravi

doi:10.1186/s13660-019-2055-1

Cited by 25 publications

(11 citation statements)

References 30 publications

(29 reference statements)

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“…These types of generalization, that is, Bivariate Szász operators is a new generalization. In this, manuscript our investigation is to generalize the Szász Durrmeyer operators based on Dunkl analogue [46] by introducing the bivariate functions. We study the bivariate properties of Szász Durrmeyer operators with the help of modulus of continuity, mixed-modulus of continuity and then find the approximation results in Peetre's K-functional, Voronovskaja type theorem and Lipschitz maximal functions for these bivariate operators.…”

Section: Conclusion and Remarksmentioning

confidence: 99%

Approximation properties of bivariate Szász Durrmeyer operators via Dunkl analogue

Rao

Heshamuddin

Shadab

et al. 2021

Filomat

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In the present article, we construct a new sequence of bivariate Sz?sz-Durrmeyer operators based on Dunkl analogue. We investigate the order of approximation with the aid of modulus of continuity in terms of well known Peetre?s K-functional, weighted approximation results, Voronovskaja type theorems and Lipschitz maximal functions. Further, we also discuss here the approximation properties of the operators in B?gel-spaces by use of mixed-modulus of continuity.

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Section: Conclusion and Remarksmentioning

confidence: 99%

Approximation properties of bivariate Szász Durrmeyer operators via Dunkl analogue

Rao

Heshamuddin

Shadab

et al. 2021

Filomat

Self Cite

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“…Lemma 2.1. [19] For e i (u) = u i , i = 0, 1, 2, 3, 4, 5, 6, the operators (2) verifies the following identities…”

mentioning

confidence: 95%

“…As a consequence of above lemma, we have Lemma 2.2. [19] The central moments for the operators (2), are…”

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confidence: 99%

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Better approximation by a Durrmeyer variant of <inline-formula><tex-math id="M1">$ \alpha- $</tex-math></inline-formula>Baskakov operators

Agrawal¹,

Singh²

2023

MFC

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<p style='text-indent:20px;'>The aim of this paper is to study some approximation properties of the Durrmeyer variant of <inline-formula><tex-math id="M2">\begin{document}$ \alpha $\end{document}</tex-math></inline-formula>-Baskakov operators <inline-formula><tex-math id="M3">\begin{document}$ M_{n,\alpha} $\end{document}</tex-math></inline-formula> proposed by Aral and Erbay [<xref ref-type="bibr" rid="b3">3</xref>]. We study the error in the approximation by these operators in terms of the Lipschitz type maximal function and the order of approximation for these operators by means of the Ditzian-Totik modulus of smoothness. The quantitative Voronovskaja and Gr<inline-formula><tex-math id="M4">\begin{document}$ \ddot{u} $\end{document}</tex-math></inline-formula>ss Voronovskaja type theorems are also established. Next, we modify these operators in order to preserve the test functions <inline-formula><tex-math id="M5">\begin{document}$ e_0 $\end{document}</tex-math></inline-formula> and <inline-formula><tex-math id="M6">\begin{document}$ e_2 $\end{document}</tex-math></inline-formula> and show that the modified operators give a better rate of convergence. Finally, we present some graphs to illustrate the convergence behaviour of the operators <inline-formula><tex-math id="M7">\begin{document}$ M_{n,\alpha} $\end{document}</tex-math></inline-formula> and show the comparison of its rate of approximation vis-a-vis the modified operators.</p>

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“…For example, Mohiuddine et al [26] introduced the family of α-Bernstein-Kantorovich operators and associated bivariate form and demonstrated the results regarding the rate of convergence via Peetre's K -functional together with modulus of continuity. In addition to this, the Stancu type α-Bernstein-Kantorovich, α-Baskakov and their Kantorovich form, and α-Baskakov-Durrmeyer operators were analyzed by Mohiuddine and Özger [32], Aral et al [6,20], and Mohiuddine et al [31], respectively, and for other blending type operators, see [23,27,36]. Some other modifications of Bernstein operators have been studied in [2,15,16,25,33,34,37,42].…”

Section: Introductionmentioning

confidence: 99%

Approximation by bivariate generalized Bernstein–Schurer operators and associated GBS operators

Mohiuddine

2020

Adv Differ Equ

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We construct the bivariate form of Bernstein–Schurer operators based on parameter α. We establish the Voronovskaja-type theorem and give an estimate of the order of approximation with the help of Peetre’s K-functional of our newly defined operators. Moreover, we define the associated generalized Boolean sum (shortly, GBS) operators and estimate the rate of convergence by means of mixed modulus of smoothness. Finally, the order of approximation for Bögel differentiable function of our GBS operators is presented.

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Approximation on parametric extension of Baskakov–Durrmeyer operators on weighted spaces

Abstract: In the present manuscript, we define a non-negative parametric variant of Baskakov-Durrmeyer operators to study the convergence of Lebesgue measurable functions and introduce these as α-Baskakov-Durrmeyer operators. We study the uniform convergence of these operators in weighted spaces.

Cited by 25 publications

References 30 publications

Approximation properties of bivariate Szász Durrmeyer operators via Dunkl analogue

Approximation properties of bivariate Szász Durrmeyer operators via Dunkl analogue

Better approximation by a Durrmeyer variant of <inline-formula><tex-math id="M1">$ \alpha- $</tex-math></inline-formula>Baskakov operators

Approximation by bivariate generalized Bernstein–Schurer operators and associated GBS operators

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