Rate of convergence of Stancu type modified $ q $-Gamma operators for functions with derivatives of bounded variation

Harun Karsli; Purshottam Narain Agrawal

doi:10.3934/mfc.2022002

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2022. Doi: 10.3934/mfc.2022002

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Rate of convergence of Stancu type modified $ q $-Gamma operators for functions with derivatives of bounded variation

Harun Karsli^1, and
Purshottam Narain Agrawal^2, ,

1.
Abant Izzet Baysal University, Faculty of Science and Arts, Department of Mathematics, 14280 Bolu, Turkey
2.
Department of Mathematics, Indian Institute of Technology Roorkee, Roorkee-247667, India

^* Corresponding author: Purshottam Narain Agrawal
^* Corresponding author: Purshottam Narain Agrawal

Received: August 31, 2021

Revised: December 31, 2021

Early access: February 2022

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Abstract

Recently, Karsli [15] estimated the convergence rate of the $ q $-Bernstein-Durrmeyer operators for functions whose $ q $-derivatives are of bounded variation on the interval $ [0, 1] $. Inspired by this study, in the present paper we deal with the convergence rate of a $ q $- analogue of the Stancu type modified Gamma operators, defined by Karsli et al. [17], for the functions $ \varphi $ whose $ q $-derivatives are of bounded variation on the interval $ [0, \infty ). $ We present the approximation degree for the operator $ \left( { \mathfrak{S}}_{n, \ell, q}^{(\alpha , \beta )} { \varphi}\right)(\mathfrak{z}) $ at those points $ \mathfrak{z} $ at which the one sided q-derivatives$ {D}_{q}^{+}{ \varphi(\mathfrak{z})\; and\; D} _{q}^{-}{ \varphi(\mathfrak{z})} $ exist.

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Mathematics Subject Classification: 41A25, 41A35.

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