EXPONENTIAL GENERALIZED DISTRIBUTIONS

Abstract

In this paper we generalize the Fourier transform from the space of tempered distributions to a bigger space called exponential generalized distributions. For that purpose we replace the Schwartz space S by a smaller space X0 of smooth functions such that, among other properties, decay at infinity faster than any exponential. The construction of X0 is such that this space of test functions is closed for derivatives, for Fourier transform and for translations. We equip X0 with an appropriate locally convex topology and we study it’s dual X’0; we call X0 0 the space of expo nential generalized distributions. The space X0 0 contains all the Schwartz tempered distributions, is closed for derivatives, and both, translations and Fourier transform, are vector and topological automorphisms in X0 0. As non trivial examples of elements in X0 0, we show that some multipole series appearing in physics are convergent in this space.