Paley-Wiener theorem for cones in locally compact Abelian groups
type: Item
instance of: scholarly article
title: Paley-Wiener theorem for cones in locally compact Abelian groups
zbMATH Open document ID: 0854.43011
author: A. R. Mirotin
published in: Russian Mathematics
publication date: 1996-05-12T00:00:00Z
review text: Let \(G\) be a locally compact Abelian group, \(X\) its dual group. Let \({\mathcal S}\) be a closed generating semigroup in \(G\) containing the identity element of \(G\) and such that its interior is dense in \({\mathcal S}\). A semicharacter of \({\mathcal S}\) is a nontrivial continuous homomorphism of \({\mathcal S}\) into the multiplicative semigroup of complex numbers \(z\) with \(|z |\leq 1\). Let \(\widehat {\mathcal S}\) denote the semigroup of all semicharacters. Let \(\widehat {\mathcal S}^0 = \widehat {\mathcal S} \backslash X\). In the paper under review a Hardy space \(H^2 (\widehat {\mathcal S}^0)\) and a Laplace transform of functions in \(L^2 ({\mathcal S})\) are defined and the following theorem (of Paley-Wiener type) is proved: if \({\mathcal S}\) satisfies some convexity condition (in this case \({\mathcal S}\) is called a cone), then the Laplace transform is an isometric isomorphism of the spaces \(L^2 ({\mathcal S})\) and \(H^2 (\widehat{\mathcal S}^0)\).
Mathematics Subject Classification ID: 43A70
Mathematics Subject Classification ID: 44A10
zbMATH DE Number: 857492
zbMATH Keywords: Paley-Wiener theorem
zbMATH Keywords: locally compact Abelian
zbMATH Keywords: dual group
zbMATH Keywords: semicharacter
zbMATH Keywords: Hardy space
zbMATH Keywords: cone
zbMATH Keywords: Laplace transform
reviewed by: V. F. Molchanov
MaRDI profile type: Publication
Recommended article: Paley-Wiener theorems on compact topological groups
Recommended article: On Wiener's lemma for locally compact abelian groups
Recommended article: Q5206769
Recommended article: A topological Paley-Wiener property for locally compact groups
Recommended article: Q3380994
Recommended article: Q3345251
Recommended article: Paley-Wiener type theorem for certain semidirect product groups
Recommended article: The Paley-Wiener theorem for certain nilpotent Lie groupsJean
Recommended article: On Paley-Wiener and Hardy theorems for \(NA\) groups
Recommended article: Q4885794
description: scientific article; zbMATH DE number 857492