Semigroup structural form for Bernstein-type operators and its applications

In this note, we provide a semigroup viewpoint for Bernstein-type operators. A main ingredient of our approach is to obtain some explicit semigroup representation formulae, which, as simple forms, are proposed for the fist time. As direct applications, we improve the corresponding results of De La Cal and Luquin (1992; 1994). Moreover, some kinds of Voronovskaja-type results are made by using general semigroup expansion. In the end, we discuss corresponding generalizations of integral operators with some examples of hybrid Durrmeyer type operators.