On the Pre-Bézout Property of Wiener Algebras on the Disc and the Half-Plane

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New Zealand Journal of Mathematics

Vol. 38, (2008), Pages 45-55

R. Mortini

Université Paul Verlaine - Metz

LMAM et Département de Mathématiques

Ile du Saulcy

F-57045 METZ

FRANCE

mailto:mortini@univ-metz.fr

A. Sasane

Mathematics Department

London School of Economics

Houghton Street

London WC2A 2AE

UNITED KINGDOM

mailto:A.J.Sasane@lse.ac.uk

Abstract Let $\mathbb{D}$ denote the open unit disk { $z$ $\in$ $\mathbb{C}$ | | $z$ | < 1 }, and $\mathbb{C}_+$ denote the closed right half-plane { $s$ $\in$ $\mathbb{C}$ | Re( $s$ ) $\ge 0$ }.

(1) Let $W +$ $\mathbb{(D)}$ be the Wiener algebra of the disc, that is the set of all absolutely convergent Taylor series in the open unit disk $\mathbb{D}$ , with pointwise operations.

(2) Let $W +$ $\mathbb{(C_+)}$ be the set of all functions defined in the right half-plane $\mathbb{C_+}$ that differ from the Laplace transform of a function $f a$ $\in$ $L 1$ $(0,\infty)$ by a constant. Equipped with pointwise operations, $W +$ $\mathbb{(C_+)}$ forms a ring.