New strips of convergence for Dirichlet series
type: Item
instance of: scholarly article
title: New strips of convergence for Dirichlet series
zbMATH Open document ID: 1207.30001
DOI: 10.5565/PUBLMAT_54210_05
author: Andreas Defant
author: Domingo García
author: Manuel Maestre
published in: Publicacions Matemàtiques
publication date: 2010-08-13T00:00:00Z
full work available at URL: https://projecteuclid.org/euclid.pm/1277731538
review text: The authors study the interplay of the theory of classical Dirichlet series in one complex variable with recent developments on monomial expansions of holomorphic functions in infinitely many variables (from the abstract). More detailed: For a Dirichlet series \(\sum_1^\infty \frac{a_n}{n^s}\) the abscissa of absolute (resp. uniform) convergence is \[ \begin{matrix} \sigma_a\\ \sigma_u\end{matrix} = \inf_{r\in\mathbb R} \left\{ \sum_1^\infty \frac{a_n}{n^s}\quad\begin{matrix}\text{absolutely}\\ \text{uniformly}\end{matrix}\text{ convergent on }\Re(s) \geq r\right\}. \] H. Bohr was interested in \(T\), the maximum of the differences \(\sigma_a - \sigma_u\) for all Dirichlet series. The Bohr--Bohnenblust--Hille theorem states that \(T=\frac12\). The authors' main aim is to extend the Bohr--Bohnenblust--Hille theorem from \(H_\infty\big(B_{\ell_\infty}\big)\) to any set \(\mathcal F\big(\ell_w \cap \mathbb D^{\mathbb N}\big)\) of holomorphic functions satisfying property (T): Given \(1\leq w\leq\infty\), a set of holomorphic functions \(\mathcal F\big(\ell_w \cap \mathbb D^{\mathbb N}\big)\) has property (T), if it remains stable under composition with diagonal operators of norm \(\leq 1\) [we do not repeat the definition of ``stable'', which is given at the beginning of \S\ 2], and if it contains the continuous polynomials on \(\ell_w\) as well as all bounded holomorphic functions on \(\ell_w \cap \mathbb D^{\mathbb N}\). The authors' main result is (Theorem 3.2): Given \(1\leq w\leq \infty\) and a set \(\mathcal F\big(\ell_w \cap \mathbb D^{\mathbb N}\big)\) of holomorphic functions containing the polynomials on \(\ell_w\), then \[ S\Big(\mathcal F\big(\ell_w \cap \mathbb D^{\mathbb N}\big)\Big) = \begin{cases} 1 & \text{if \(1\leq w<2\)}, \\ \left(\frac12 +\frac1w\right)^{-1} & \text{if \( 2\leq w \leq \infty\)},\end{cases} \] and if, in addition, \(\mathcal F\big(\ell_w \cap \mathbb D^{\mathbb N}\big)\) satisfies property (T), then \[ T\Big(\mathcal F\big(\ell_w \c…
Mathematics Subject Classification ID: 30B50
Mathematics Subject Classification ID: 32A05
Mathematics Subject Classification ID: 32A10
zbMATH DE Number: 5770005
zbMATH Keywords: abscissa of absolute convergence
zbMATH Keywords: abscissa of uniform convergence
zbMATH Keywords: Dirichlet series
zbMATH Keywords: formal power series with infinitely many variables
zbMATH Keywords: Reinhardt domains
zbMATH Keywords: infinite dimensional holomorphy
reviewed by: Wolfgang Schwarz
MaRDI profile type: Publication
description: scientific article; zbMATH DE number 5770005