EXPONENTIAL GENERALIZED DISTRIBUTIONS

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