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

### 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

A. Sasane

Mathematics Department

London School of Economics

Houghton Street

London WC2A 2AE

UNITED KINGDOM

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 fa $\in$ L1$(0,\infty)$ by a constant. Equipped with pointwise operations, W + $\mathbb{(C_+)}$forms a ring.

We show that the rings W + $\mathbb{(D)}$ and W + $\mathbb{(C_+)}$ are pre-Bézout rings.

Keywords pre-Bézout ring, Wiener algebra

Classification (MSC2000) Primary: 30H05; Secondary: 42A99, 93D15