Noncoherence of a Causal Wiener Algebra Used in Control Theory

Abstract EN

Let ℂ≥0:={s∈ℂ∣Re (s)≥0}, and let ?+ 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

(Here ⋅^ denotes the Laplace transform.) It is shown that the ring ?+ is not coherent, answering a question of Alban Quadrat.

In fact, we present two principal ideals in the domain ?+ whose intersection is not finitely generated.