International
Tables for Crystallography Volume B Reciprocal space Edited by U. Shmueli © International Union of Crystallography 2010 
International Tables for Crystallography (2010). Vol. B, ch. 1.3, pp. 3738

The duality established in Sections 1.3.2.4.2.8 and 1.3.2.4.2.9 between the local differentiability of a function and the rate of decrease at infinity of its Fourier transform prompts one to consider the space of functions f on which are infinitely differentiable and all of whose derivatives are rapidly decreasing, so that for all multiindices k and pThe product of by any polynomial over is still in ( is an algebra over the ring of polynomials). Furthermore, is invariant under translations and differentiation.
If , then its transforms and are
hence and are in : is invariant under and .
Since and , all properties of and already encountered above are enjoyed by functions of , with all restrictions on differentiability and/or integrability lifted. For instance, given two functions f and g in , then both fg and are in (which was not the case with nor with ) so that the reciprocity theorem inherited from allows one to state the reverse of the convolution theorem first established in :