  A Hilbert space approach to the classical Fantappie transform, based on the
concept of Gel'fand triples of locally convex spaces, leads to a novel proof of
Martineau-Aizenberg duality theorem. A study of Fantappie transforms of
positive measures on the unit ball in $\C^n$ relates ideas of realization
theory of multivariate linear systems, locally convex duality and
pluripotential theory. This is applied to obtain von Neumann type estimates on
the joint numerical range of tuples of Hilbert space operators.
