On Weighted $L^p$-Approximation by Weighted Bernstein-Durrmeyer Operators

Abstract

In the present paper, we establish direct and converse theorems for weighted Bernstein-Durrmeyer operators under weighted $L^p$−norm with Jacobi weight $w(x) = x^{\alpha}(1−x)^{\beta}$. All the results involved have no restriction $\alpha$, $\beta<1-\frac{1}{p}$, which indicates that the weighted Bernstein-Durrmeyer operators have some better approximation properties than the usual Bernstein-Durrmeyer operators.