  The notion of Fourier transformation is described from an algebraic
perspective that lends itself to applications in Symbolic Computation. We build
the algebraic structures on the basis of a given Heisenberg group (in the
general sense of nilquadratic groups enjoying a splitting property); this
includes in particular the whole gamut of Pontryagin duality. The free objects
in the corresponding categories are determined, and various examples are given.
As a first step towards Symbolic Computation, we study two constructive
examples in some detail -- the Gaussians (with and without polynomial factors)
and the hyperbolic secant algebra.
