Paley-Wiener theorem for cones in locally compact Abelian groups
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 \(H^2 (\widehat{\mathcal S}^0)\).
scientific article; zbMATH DE number 857492