A generalised beta integral and the limit of the Bernstein--Durrmeyer operator with Jacobi weights

Abstract

We give a generalisation of the multivariate beta integral. This is used to show that the (multivariate) Bernstein--Durrmeyer operator for a Jacobi weight has a limit as the weight becomes singular. The limit is an operator previously studied by Goodman and Sharma. From the elementary proof given, it follows that this operator inherits many properties of the Bernstein--Durrmeyer operator in a natural way. In particular, we determine its eigenstructure and give a differentiation formula for it.