Hankel 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 p ( ℝ q ) , q ≥ 1 . We will replace the ϕ q , q ≥ 1 with test functions having Hankel transforms. This space is a natural generalization of a classical Fock space as seen by expanding functionals having abstract Taylor Series. The particular coefficients of such series are multilinear functionals having distributions as their domain. Convergence requirements set forth are somewhat in the spirit of ultra differentiable functions and ultra distribution theory. The Hankel transform oftentimes implemented in Cauchy problems will be introduced into this setting. A theorem will be proven relating the convergence of the transform to the inductive limit parameter, s , which sweeps out a scale of generalized Fock spaces.