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Abstract and Applied Analysis
Volume 2008 (2008), Article ID 459310, 13 pages
http://dx.doi.org/10.1155/2008/459310
Research Article

Noncoherence of a Causal Wiener Algebra Used in Control Theory

Mathematics Department, London School of Economics, Houghton Street, London WC2A 2AE, UK

Received 18 March 2008; Accepted 13 June 2008

Academic Editor: Ülle Kotta

Copyright © 2008 Amol Sasane. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

Abstract

Let β„‚ β‰₯ 0 ∢ = { 𝑠 ∈ β„‚ ∣ R e ( 𝑠 ) β‰₯ 0 } , and let 𝒲 + denote the ring of all functions 𝑓 ∢ β„‚ β‰₯ 0 β†’ β„‚ such that 𝑓 ( 𝑠 ) = 𝑓 π‘Ž ( 𝑠 ) + βˆ‘ ∞ π‘˜ = 0 𝑓 π‘˜ 𝑒 βˆ’ 𝑠 𝑑 π‘˜ ( 𝑠 ∈ β„‚ β‰₯ 0 ) , where 𝑓 π‘Ž ∈ 𝐿 1 ( 0 , ∞ ) , ( 𝑓 π‘˜ ) π‘˜ β‰₯ 0 ∈ β„“ 1 , and 0 = 𝑑 0 < 𝑑 1 < 𝑑 2 < β‹― equipped with pointwise operations. (Here 0 π‘₯ 0 0 0 5 𝑒 β‹… 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.