American Institute of Mathematical Sciences

doi: 10.3934/mfc.2020023

On approximation to discrete q-derivatives of functions via q-Bernstein-Schurer operators

 Bolu Abant Izzet Baysal University, Faculty of Science and Arts, Department of Mathematics, 14030, Golkoy-Bolu, Turkey

* Corresponding author: Harun Karsli

Received  April 2020 Revised  September 2020 Published  November 2020

In the present paper, we shall investigate the pointwise approximation properties of the $q-$analogue of the Bernstein-Schurer operators and estimate the rate of pointwise convergence of these operators to the functions $f$ whose $q-$derivatives are bounded variation on the interval $[0,1+p].$ We give an estimate for the rate of convergence of the operator $\left( B_{n,p,q}f\right)$ at those points $x$ at which the one sided $q-$derivatives $D_{q}^{+}f(x)$ and $D_{q}^{-}f(x)$ exist. We shall also prove that the operators $\left( B_{n,p,q}f\right) (x)$ converge to the limit $f(x)$. As a continuation of the very recent and initial study of the author deals with the pointwise approximation of the $q-$Bernstein Durrmeyer operators [12] at those points $x$ at which the one sided $q-$derivatives $D_{q}^{+}f(x)$ and $D_{q}^{-}f(x)$ exist, this study provides (or presents) a forward work on the approximation of $q$-analogue of the Schurer type operators in the space of $D_{q}BV$.

Citation: Harun Karsli. On approximation to discrete q-derivatives of functions via q-Bernstein-Schurer operators. Mathematical Foundations of Computing, doi: 10.3934/mfc.2020023
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