Asymptotic Approximation by Bernstein-Durrmeyer Operators and their Derivatives

Abstract

The concern of this paper is the study of local approximation properties of the Bernstein-Durrmeyer operators Mn. We derive the complete asymptotic expansion of the operators Mn and their derivatives as n tends to infinity. It turnsout that the appropriate representation is a series of reciprocal factorials. All coefficients are calculated explicitly in a very concise form. Our main theorem contains several earlier partial results as special cases. Moreover, it may beuseful for further investigations on Bernstein-Durrmeyer operators. Finally, we obtain a Voronovskaja-type formula for the simultaneous approximation by linear combinations of the Mn.