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 X&#8242;0 the space of exponential generalized distributions. The space X&#8242;0 contains all the Schwartz tempered distributions, is closed for derivatives, and both, translations and Fourier transform, are vector and topological automorphisms in X&#8242;0. As non trivial examples of elements in X&#8242;0, we show that some multipole series appearing in physics are convergent in this space.</p