International Journal of Mathematics and Mathematical Sciences
Volume 17 (1994), Issue 2, Pages 259-272
doi:10.1155/S0161171294000384
Hankel transforms in generalized Fock spaces
Department of Mathematical Sciences, Box 2014, Oliver Hall, 1015 W. Main Street, Virginia Commonwealth University, Richmond 23284-2014, Virginia, USA
Received 21 July 1992; Revised 6 April 1993
Copyright © 1994 John Schmeelk. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.
Abstract
A classical Fock space consists of functions of the form,ϕ↔(ϕ0,ϕ1,…,ϕq),where ϕ0∈ℂ and ϕq∈Lp(ℝq), q≥1. We will replace the ϕq, 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.