Paley-Wiener theorems on compact topological groups
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.
scientific article; zbMATH DE number 40555