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. 49

The usual presentation of this duality is not in terms of lattice distributions, but of periodic distributions obtained by convolving them with a motif.
Given , let us form , then decimate its transform by keeping only its values at the points of the coarser lattice ; as a result, is replaced by , and the reverse transform then yieldswhich is the cosetaveraged version of the original . The converse situation is analogous to that of Shannon's sampling theorem. Let a function whose transform has compact support be sampled as at the nodes of . Thenis periodic with period lattice . If the sampling lattice is decimated to , the inverse transform becomeshence becomes periodized more finely by averaging over the cosets of . With this finer periodization, the various copies of Supp Φ may start to overlap (a phenomenon called `aliasing'), indicating that decimation has produced too coarse a sampling of ϕ.