Operational calculus and generalized functions

Abstract It has recently been shown that operators of a certain type can be used to generalize the concept of a function and to extend the Laplace-transform calculus. A further study of these operators is now undertaken, and some new characterizations of them are obtained. In a preliminary discussion, the notion of an operational calculus is examined from an axiomatic point of view, and a general definition is formulated; systems of generalized functions are also considered axiomatically, and are shown to be necessary for the extension, under mild conditions, of differential operators on functions of a real variable.