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



A. Sasane

Mathematics Department

London School of Economics

Houghton Street

London WC2A 2AE



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

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