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

Zbl 0677.46039
Full Text:

References:

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