Three Theorems of Paley and Wiener for Locally Compact Abelian Groups

The representation theorems of Paley and Wiener concerning analytic functions on a vertical strip, analytic functions on the right halfplane, and entire functions of exponential type are generalized within the context of abstract harmonic analysis. The holomorphic Fourier transform is generalized as in Liepins (7), but a different, vector-valued abstraction of analyticity is employed. The proofs of these results use standard facts from harmonic analysis, integration theory, and the theory of vector-valued analytic functions, but are otherwise elementary; in particular, no use is ma.de of structure theorems for locally compact abelian groups. Finally, the appendix contains several analogous results on maxima 1 ideal spaces of related convolution algebras.