Limits of certain iterates of Post-Widder operators

The paper is a study of the limiting behaviour of the [n t]-th iterates of the well-known Post-Widder operatorsL_{n, x} used in the real inversion of the Laplace transform. It is shown that the limiting operators constitute a semigroup T_t;t0 of class (C₀) on a family C_,; , >0 of Banach spaces. Applications of the semigroup structure lead to a pointwise saturation theorem forL_{n, x} and a characterization of convex functions inC_, through an inequality involving the action ofL_{n, x}.