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. 4243

The content of this section is perhaps the central result in the relation between Fourier theory and crystallography (Section 1.3.4.2.1.1).
Let with r defined as in Section 1.3.2.6.2. Then , hence , so that : periodic distributions are tempered, hence have a Fourier transform. The convolution theorem (Section 1.3.2.5.8) is applicable, giving:and similarly for .
It is readily shown that Q is tempered and periodic, so that , while the periodicity of r implies thatSince the first factors have single isolated zeros at in , (see Section 1.3.2.3.9.4) and hence by periodicity ; convoluting with shows that . Thus we have the fundamental result: so thati.e., according to Section 1.3.2.3.9.3,
The righthand side is a weighted lattice distribution, whose nodes are weighted by the sample values of the transform of the motif at those nodes. Since , the latter values may be writtenBy the structure theorem for distributions with compact support (Section 1.3.2.3.9.7), is a derivative of finite order of a continuous function; therefore, from Section 1.3.2.4.2.8 and Section 1.3.2.5.8, grows at most polynomially as (see also Section 1.3.2.6.10.3 about this property). Conversely, let be a weighted lattice distribution such that the weights grow at most polynomially as . Then W is a tempered distribution, whose Fourier cotransform is periodic. If T is now written as for some , then by the reciprocity theoremAlthough the choice of is not unique, and need not yield back the same motif as may have been used to build T initially, different choices of will lead to the same coefficients because of the periodicity of .
The Fourier transformation thus establishes a duality between periodic distributions and weighted lattice distributions. The pair of relationsare referred to as the Fourier analysis and the Fourier synthesis of T, respectively (there is a discrepancy between this terminology and the crystallographic one, see Section 1.3.4.2.1.1). In other words, any periodic distribution may be represented by a Fourier series (ii), whose coefficients are calculated by (i). The convergence of (ii) towards T in will be investigated later (Section 1.3.2.6.10).