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, p. 44

Let be a distribution with compact support (the `motif'). Its Fourier transform is analytic (Section 1.3.2.5.4) and may thus be used as a multiplier.
We may rephrase the preceding results as follows:
Thus the Fourier transformation establishes a duality between the periodization of a distribution by a period lattice Λ and the sampling of its transform at the nodes of lattice reciprocal to Λ. This is a particular instance of the convolution theorem of Section 1.3.2.5.8.
At this point it is traditional to break the symmetry between and which distribution theory has enabled us to preserve even in the presence of periodicity, and to perform two distinct identifications: