Summary: 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 \(\mathcal S\) by a smaller space \({\mathcal X}_0\) of smooth functions such that, among other properties, decay at infinity is faster than any exponential. The construction of \({\mathcal X}_0\) is such that this space of test functions is closed for derivatives, for the Fourier transform and for translations. We equip \({\mathcal X}_0\) with an appropriate locally convex topology and we study its dual \({\mathcal X}'_0\), we call \({\mathcal X}'_0\) the space of exponential generalized distributions. The space \({\mathcal X}'_0\) contains all the Schwartz tempered distributions, is closed for derivatives, and both translations and the Fourier transform are vector and topological automorphisms in \({\mathcal X}'_0\). As nontrivial examples of elements in \({\mathcal X}'_0\), we show that some multipole series appearing in physics are convergent in this space.