International
Tables for Crystallography Volume B Reciprocal space Edited by U. Shmueli © International Union of Crystallography 2006 
International Tables for Crystallography (2006). Vol. B, ch. 1.3, pp. 4647

The above relations between lattices may be rewritten in terms of the corresponding lattice distributions as follows: where and are (finite) residuallattice distributions. We may incorporate the factor in (i) and into these distributions and define
Since , convolution with and has the effect of averaging the translates of a distribution under the elements (or `cosets') of the residual lattices and , respectively. This process will be called `coset averaging'. Eliminating and between (i) and (ii), and and between and , we may write: These identities show that period subdivision by convolution with (respectively ) on the one hand, and period decimation by `dilation' by on the other hand, are mutually inverse operations on and (respectively and ).