A Modification of Bernstein-Durrmeyer Operators with Jacobi Weights on the Unit Interval

Abstract

The present paper is devoted to the study of a sequence of positive linear operators, acting on the space of all continuous functions on [0, 1] as well as on some weighted spaces of integrable functions on [0, 1]. These operators are, as a matter of fact, a generalization of the Bernstein-Durrmeyer operators with Jacobi weights. In particular, we present qualitative and approximation properties of these operators, also providing estimates of the rate of convergence. Moreover, by means of their asymptotic formula, we compare our operators with the Bernstein-Durrmeyer ones and a suitable modification of theirs, showing that, in suitable intervals, they provide a lower approximating error estimate.