Date of this Version
Bulletin Of The American Mathematical Society Volume 77, Number 3, May 1971
The resulting "Fourier transformation" µ—µ^ 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 [ l l ] introduced a direct generalization to this setting of the theory of Schwartz , 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.