Characterization of Noetherian rings of formal power series with controlled growth. Application to spectral synthesis (Q1385198)

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.