Abstract and Applied Analysis
Volume 2008 (2008), Article ID 459310, 13 pages
doi:10.1155/2008/459310

Noncoherence of a causal Wiener algebra used in control theory

Abstract

Let ℂ≥0:={s∈ℂ∣Re⁡(s)≥0}, and let 𝒲+ denote the ring of all functions f:ℂ≥0→ℂ such that f(s)=fa(s)+∑k=0∞fke−stk (s∈ℂ≥0), where fa∈L1(0,∞), (fk)k≥0∈ℓ1, and  0=t0<t1<t2<⋯ equipped with pointwise operations. (Here ⋅^ denotes the Laplace transform.) It is shown that the ring 𝒲+ is not coherent, answering a question of Alban Quadrat. In fact, we present two principal ideals in the domain 𝒲+ whose intersection is not finitely generated.