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

Throughout this section, `periodic' will mean `periodic'.
Let , and let [s] denote the largest integer . For , let be the unique vector with . If , then if and only if . The image of the map is thus modulo , or .
If f is a periodic function over , then implies ; we may thus define a function over by putting for any such that . Conversely, if is a function over , then we may define a function f over by putting , and f will be periodic. Periodic functions over may thus be identified with functions over , and this identification preserves the notions of convergence, local summability and differentiability.
Given , we may definesince the sum only contains finitely many nonzero terms; ϕ is periodic, and . Conversely, if we may define periodic by , and by putting with ψ constructed as above.
By transposition, a distribution defines a unique periodic distribution by ; conversely, periodic defines uniquely by .
We may therefore identify periodic distributions over with distributions over . We will, however, use mostly the former presentation, as it is more closely related to the crystallographer's perception of periodicity (see Section 1.3.4.1).