The Fourier--Borel transform between spaces of entire functions of a given type and order
type: Item
instance of: scholarly article
title: The Fourier--Borel transform between spaces of entire functions of a given type and order
zbMATH Open document ID: 1152.46033
DOI: 10.4171/PM/1813
author: V. V. Fávaro
published in: Portugaliae Mathematica. Nova Série
publication date: 2008-07-02T00:00:00Z
full work available at URL: http://www.ems-ph.org/journals/show_pdf.php
review text: Given a Banach space \(E\) and a positive integer \(m\), denote by \({\mathcal P}_ {(s,m(r,q))}(^mE)\) (resp., \({\mathcal P}_{\tilde N,(s,(r,q))}(^mE))\) the space of all \((s,m(r,q))\)-summing (resp., \((s;(r,q))\)-quasi-nuclear) \(m\)-homogeneous polynomials on \(E\). For \(\rho>0\) and \(k\geq 1\), the author introduces the Banach spaces \({\mathcal B}^k_{(s,m(r;q)),\rho}(E)\) (resp., \({\mathcal B}^k_{\tilde N,(s;(r;q)), \rho}(E)\)), consisting of all holomorphic function \(f\) on \(E\) such that \(\hat d^mf(0)\) belongs to \({\mathcal P}_{(s,m(r,q))}(^mE)\) (resp., \({\mathcal P}_{\tilde N,(s,(r,q))} (^mE)\)) and \[ \sum_{j=0}^\infty\rho^{-j}\left({j\over ke}\right)^{j/k}\left\|{\hat d^mf(0)\over j!}\right\|_{(s,m(r,q))}<\infty, \] resp., \[ \sum_{j=0}^\infty\rho^{-j}\left({j\over ke}\right)^{j/k}\left\|{\hat d^mf(0)\over j!}\right\|_{\tilde N,(s,m(r,q))}<\infty. \] Taking the inductive limit over all \(\rho<A\) (resp., projective limit over all \(\rho\geq A\)), he obtains the DF-spaces (resp., Fréchet spaces) \(\text{Exp}^k_{(s,m(r;q)),A}(E)\) and \(\text{Exp}^k_{\tilde N,(s,(r,q)),A}(E)\) (resp., \(\text{Exp}^k_{(s,m(r;q)),0,A}(E)\) and \(\text{Exp}^k_{\tilde N,(s,(r,q)),0,A}(E)\)). The main result of this paper is that when \(E'\) has the \(\lambda\)-approximation property, then the mapping \(T\mapsto FT\) defined by \(FT(\phi)=T(e^\phi)\) establishes an algebraic isomorphism between the spaces \([\text{Exp}^k_{{\tilde N},(s,(r,q)),A}(E)]'\) and \(\text{Exp}^{k'}_{(s',m(r';q'))0, \theta(k)A^{-1}}(E')\) and between \([\text{Exp}^k_{{\tilde N},(s,(r,q)),0, A}(E)]'\) and \(\text{Exp}^{k'}_{(s',m(r';q')),\theta(k)A^{-1}}(E')\). Here, \(\theta(k)={k\over (k-1)^{(k-1)/k}}\). Using a description of the bounded sets in these spaces, the author shows that the first of these algebraic isomorphisms is, in fact, topological when the dual is endowed with the strong topology.
reviewed by: Christopher Boyd
Mathematics Subject Classification ID: 46G20
Mathematics Subject Classification ID: 46G25
zbMATH DE Number: 5296713
zbMATH Keywords: holomorphic function
zbMATH Keywords: exponential type
zbMATH Keywords: absolute summing polynomials
zbMATH Keywords: quasi-nuclear polynomial
zbMATH Keywords: Fourier-Borel Transform
MaRDI profile type: Publication
cites work: Q5183990
cites work: Q5654656
cites work: Équations différentielles d'ordre infini
cites work: Q3684655
description: scientific article; zbMATH DE number 5296713