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 nat...

Here we provide a unifying treatment of the convergence general form sampling type operators, given by so-called Durrmeyer series. In particular pointwise and uniform theorem on $\mathbb{R}$, in this context also furnish quantitative estimate for order approximation, using modulus continuity function to be approximated. Then obtain modular setting Orlicz spaces $L^\varphi(\mathbb{R})$. From lat...