Some properties of q-Bernstein-Durrmeyer operators

Abstract

In the present paper we shall investigate the pointwise approximation properties of the q analogue of the Bernstein-Durrmeyer operators and estimate the rate of pointwise convergence of these operators to the functions $f$ whose q-derivatives are bounded variation on the interval $[0,1]$. We give an estimate for the rate of convergence of the operator $\left( L_{n,q}f \right)$ at those points $x$ at which the one sided q-derivatives $ D_{q}^{+}f(x),D_{q}^{-} f(x)$ exist. We shall also prove that the operators $L_{n,q}f$ converges to the limit $f(x).$ To the best of my knowledge, the present study will be the first study on the approximation of q- operators in the space of $D_{q}BV$.