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Noncoherence of a causal Wiener algebra used in control theory. (English) Zbl 1159.46032

Let \(R\) be a commutative ring with identity element 1 and let \(R^m=R\times\dots\times R\) (\(m\) times). Suppose that \(f=(f_1, f_2,\dots f_m)\in R^m\).
(1)
An element \(g=(g_1, g_2,\dots g_m)\in R^m\) is called a relation on \(f\) if
\[ g_1f_1+\dots + g_mf_m=0. \]
(2)
Let \(f^{\bot}\) denote the set of all relations on \(f\in R^m\). (Then \(f^{\bot}\) is an \(R\)-submodule of the \(R\)-module \(R^m\).)
(3)
The ring \(R\) is called coherent if for all \(m\in\mathbb N\) and all \(f\in R^m\), \(f^{\bot}\) is finitely generated, that is, there exists \(d\in\mathbb N\) and there exists \(g_j\in f^{\bot}\), \(j\in\{1,\dots, d\}\), such that for all \(g\in f^{\bot}\), there exist \(r_j\in R\), \(j\in\{1,\dots, d\}\), such that \(g=r_1g_1+\dots + r_dg_d.\)
Let \(\mathbb C_{\geq 0}=\{s\in\mathbb C: \operatorname{Re}(s)\geq 0\}\) and let \({\mathcal W}^{+}\) denote the Banach algebra of those functions \(f:\mathbb C_{\geq 0}\to \mathbb C\) such that
\[ f(s)=\widehat{f_a}(s)+\sum_{k=0}^\infty f_ke^{-st_k}, \quad s\in \mathbb C_{\geq 0}, \]
where \(f_a: (0,\infty)\to\mathbb C\), \(f_a\in L^1(0,\infty)\), for all \(k\in\mathbb N\), \(f_k\in\mathbb C\), \(\{f_k\}_{k=0}^\infty\in l^1\), for all \(k\in\mathbb N\), \(t_k\in\mathbb C\), \(0= t_0<t_1<t_2<\dots\), equipped with the pointwise operations and the norm
\[ \|f\|_{\mathcal W}^{+}= \|f_a\|_{L^1}+\|\{f_k\}_{k=0}^\infty\|_{l^1}. \]
Here, \(\widehat{f_a}\) denotes the Laplace transform of \(f_a\), given by
\[ \widehat{f_a}(s)=\int_0^\infty e^{-st}f_a(t)\,dt,\quad s\in\mathbb C_{\geq 0}. \]
In this paper, it is proved, using ideas of R. Mortini and M. von Renteln [J. Aust. Math. Soc., Ser. A 46, No. 2, 220–228 (1989; Zbl 0677.46039)], that the ring \({\mathcal W}^+\) is not coherent.

MSC:

46J15 Banach algebras of differentiable or analytic functions, \(H^p\)-spaces
30H05 Spaces of bounded analytic functions of one complex variable
13E99 Chain conditions, finiteness conditions in commutative ring theory

Citations:

Zbl 0677.46039
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References:

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