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. 42

A distribution is called periodic with period lattice (or periodic) if for all (in crystallography the period lattice is the direct lattice).
Given a distribution with compact support , then is a periodic distribution. Note that we may write , where consists of Dirac δ's at all nodes of the period lattice .
Conversely, any periodic distribution T may be written as for some . To retrieve such a `motif' from T, a function ψ will be constructed in such a way that (hence has compact support) and ; then . Indicator functions (Section 1.3.2.2) such as or cannot be used directly, since they are discontinuous; but regularized versions of them may be constructed by convolution (see Section 1.3.2.3.9.7) as , with and η such that on and outside . Then the functionhas the desired property. The sum in the denominator contains at most nonzero terms at any given point x and acts as a smoothly varying `multiplicity correction'.