Characterization of Noetherian rings of formal power series with controlled growth. Application to spectral synthesis
type: Item
instance of: scholarly article
title: Characterization of Noetherian rings of formal power series with controlled growth. Application to spectral synthesis
zbMATH Open document ID: 0906.46017
DOI: 10.5565/PUBLMAT_41297_18
author: Jacques Chaumat
author: Anne-Marie Chollet
published in: Publicacions Matemàtiques
publication date: 1999-02-01T00:00:00Z
full work available at URL: https://eudml.org/doc/41305
review text: Let \(M=\{M_p\}_{p\geq 0}\) be a logarithmically convex sequence of positive numbers with \(M_0=1\) and \(\lim_{p\to\infty}M_p=\infty\). For \(C>0\), denote by \(M(n,C)\) the set of formal power series in \(n\) variables which satisfy the growth estimate \(\sup_{J\in{\mathbb N}^n} | a_J| /(C^{| J| }M_{| J| })<\infty\), and define \(BM(n)=\bigcap_{C>0}M(n,C)\) and \(CM(n)=\bigcup_{C>0}M(n,C)\). The authors show that these rings are noetherian if and only if \(\{M_p\}\) has the additional property that there exists \(A\geq 1\) such that \(M_{p+1}\leq A^{p+1}M_p\) for all \(p\geq 0\). This is done by first proving a theorem that allows division without loss of regularity. As an application of the main theorem, the authors obtain a spectral synthesis theorem for a class of ultradifferentiable functions.
reviewed by: Line Baribeau
Mathematics Subject Classification ID: 46E10
Mathematics Subject Classification ID: 46J20
Mathematics Subject Classification ID: 26E10
zbMATH DE Number: 1146203
zbMATH Keywords: Noetherian rings
zbMATH Keywords: spectral synthesis
zbMATH Keywords: ultradifferentiable classes
zbMATH Keywords: logarithmically convex sequence of positive numbers
MaRDI profile type: Publication
description: scientific article; zbMATH DE number 1146203