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

Shayne Waldron


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.

Keywords: Bernstein--Durrmeyer operator, multivariate beta integral, Jacobi polynomials

Math Review Classification: Primary 33B15, 41A10 ; Secondary 15A18, 33C45, 41A36

Length: 10 pages

Last Updated: 19 February 2002