Heaviside's operational calculus and the attempts to rigorise it

At the end of the 19^th century Oliver Heaviside developed a formal calculus of differential operators in order to solve various physical problems. The pure mathematicians of his time would not deal with this unrigorous theory, but in the 20^th century several attempts were made to rigorise Heaviside's operational calculus. These attempts can be grouped in two classes. The one leading to an explanation of the operational calculus in terms of integral transformations (Bromwich, Carson, Vander Pol, Doetsch) and the other leading to an abstract algebraic formulation (Lévy, Mikusiski). Also Schwartz's creation of the theory of distributions was very much inspired by problems in the operational calculus.