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

  • * Corresponding author: Harun Karsli

    * Corresponding author: Harun Karsli
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  • 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 $.

    Mathematics Subject Classification: Primary: 41A25; Secondary: 41A35.


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