Fourier transforms in generalized Fock spaces

A classical Fock space consists of functions of the form, Φ ↔ ( ϕ 0 , ϕ 1 , … , ϕ q , … ) , where ϕ 0 ∈ C and ϕ q ∈ L 2 ( R 3 q ) , q ≥ 1 . We will replace the ϕ q , q ≥ 1 with q -symmetric rapid descent test functions within tempered distribution theory. This space is a natural generalization of a classical Fock space as seen by expanding functionals having generalized Taylor series. The particular coefficients of such series are multilinear functionals having tempered distributions as their domain. The Fourier transform will be introduced into this setting. A theorem will be proven relating the convergence of the transform to the parameter, s , which sweeps out a scale of generalized Fock spaces.