Hankel transforms in generalized Fock spaces

John Schmeelk

doi:10.1155/s0161171294000384

Abstract

A classical Fock space consists of functions of the form, urn:x-wiley:01611712:media:ijmm356721:ijmm356721-math-0001 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.

Highlights

The test space, g,/ e (-o0,oo) consisting of continuous complex-valued function defined on:, the q-dimensional orthant, Eq {t e Rq: 0 < 7 < o0, (1 < 3’ < q)} and its dual space, are excellent candidates for examining the Hankel transform (Brychkov and Prudnikov [1], Koh [2], Pathak and Singh [3] and Zemanian [4])
The Hankel transform in this setting investigates g, spaces having test functions, e defined on a finite number of independent variables. By this we mean the independent variables of a test function, e has finitely many independent variables, belonging to Eq Our present development will indicate a process whereby the independent variables, t.r, 1 < 7 _< q, can become infinite in the sense that the dimension, q
Will be to extend the transform to our spaces defined as generalized Fock spaces

Summary

Introduction

The test space, g,/ e (-o0,oo) consisting of continuous complex-valued function defined on:, the q-dimensional orthant, Eq {t e Rq: 0 < 7 < o0, (1 < 3’ < q)} and its dual space, are excellent candidates for examining the Hankel transform (Brychkov and Prudnikov [1], Koh [2], Pathak and Singh [3] and Zemanian [4]). The Hankel transform in this setting investigates g, spaces having test functions, e defined on a finite number of independent variables. With these settings in place, we will extend the Hankel transform into inductive and projective limit spaces (Zarinov [9]).

Results

Conclusion