Paley-Wiener theorems on compact topological groups
type: Item
instance of: scholarly article
title: Paley-Wiener theorems on compact topological groups
zbMATH Open document ID: 0764.43004
DOI: 10.4171/ZAA/444
author: Hans-Friedrich Bauch
published in: Zeitschrift für Analysis und ihre Anwendungen
publication date: 1992-08-13T00:00:00Z
review text: Let \(G\) be a compact connected group with dual object \(\Sigma\), \(\Lambda:=\Hom(\mathbb{R},G)\). Let \(K=L_ 2*L_ 2\) be the Banach space of functions with absolutely converging (operator valued) Fourier series and let \(S(G)\) resp. \(ES(G)\) be the spaces of functions with rapidly decreasing resp. exponential rapidly decreasing \(\widehat f\). The author considers functions on \(G\) which are differentiable in the sense of Boseck and Czichowski, i.e. \(t\mapsto f(g\lambda(t))\) is differentiable for \(g\in G\), \(\lambda\in\Lambda\). Let \(R\) be the regular representation of \(G\) on \(K(G)\). Then (Satz 1, 2, 3) if \(f\in ES(G)\) then \(f\) is a differentiable vector, and any differentiable vector of \(R\) belongs to \(S(G)\). Moreover, \(ES(G)\) is the set of analytical vectors of \(R\) and functions of \(ES(G)\) admit a complex-analytic representation. In \S3 the author considers abelian groups and finally in \S4 the results are compared with the stronger differentiability concept of Bruhat-Maurin- Kac.
Mathematics Subject Classification ID: 43A75
Mathematics Subject Classification ID: 43A30
Mathematics Subject Classification ID: 43A25
Mathematics Subject Classification ID: 43A40
Mathematics Subject Classification ID: 22C05
zbMATH DE Number: 40555
zbMATH Keywords: compact connected group
zbMATH Keywords: dual object
zbMATH Keywords: Fourier series
zbMATH Keywords: exponential rapidly decreasing
zbMATH Keywords: regular representation
zbMATH Keywords: complex-analytic representation
zbMATH Keywords: differentiability
reviewed by: Wilfried Hazod
MaRDI profile type: Publication
description: scientific article; zbMATH DE number 40555