Fourier Transforms of Unbounded Measures

Let G be a locally compact abelian group. For a (generally unbounded) measure μ on G we shall say that μ is transformable if there is a measure μˆ on the character group Γ of G such that, for every ƒЄK(G), the space of continuous functions with compact support on G,ƒЄL2(μ) and (1) ∫Gƒ**ƒ(x) dμ(x) = ∫r|∫(γ-1)|2dμ(γ). The resulting Fourier transformation” μ→μˆ contains the classical theory and leads to generalizations of a variety of classical results, including the Plancherel theorem and the Poisson summation formula. The present work can also be regarded as a sort of theory of tempered distributions on general locally compact abelian groups. It is true that Bruhat [11] introduced a direct generalization to this setting of the theory of Schwartz [10], but, to the best of our knowledge, a detailed study of the Fourier transform has not been carried out. In a forthcoming exposition we shall describe the precise relation between the present study and the work of Bruhat