First, they can be used to verify the general theory developed in , and at the same time they provide a new kind of functional calculus, in particular they provide the possibility to study functional calculi for non-commuting unbounded operators.
In each case the Weyl calculus is reobtained, more specifically the Weyl calculus is the value of functional calculus at the identity of each affine group.
But Dana Scott developed a theory of domains--partially ordered sets of a special nature--which provides meaning for the [unkeyable]-calculus, the prime functional calculus.
We wish to match the functional calculus not by copying its constructions, but by emulating two of its attributes: It is synthetic--we build systems in it, because the structure of terms represents the structure of processes; and it is computational--its basic semantic notion is a step of computation.
He moves from his discussion of functionals to the problems of the functional calculus
and integral equations, the generalization of the analytic functions, the theory of composition and of permutable functions, and then to integro-differential equations and functional derivative equations.