Tables for
Volume B
Reciprocal space
Edited by U. Shmueli

International Tables for Crystallography (2006). Vol. B, ch. 1.3, p. 41   | 1 | 2 |

Section [{\bb Z}^{n}]-periodic distributions in [{\bb R}^{n}]

G. Bricognea

aMRC Laboratory of Molecular Biology, Hills Road, Cambridge CB2 2QH, England, and LURE, Bâtiment 209D, Université Paris-Sud, 91405 Orsay, France [{\bb Z}^{n}]-periodic distributions in [{\bb R}^{n}]

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A distribution [T \in {\scr D}\,' ({\bb R}^{n})] is called periodic with period lattice [{\bb Z}^{n}] (or [{\bb Z}^{n}]-periodic) if [\tau_{\bf m} T = T] for all [{\bf m} \in {\bb Z}^{n}] (in crystallography the period lattice is the direct lattice).

Given a distribution with compact support [T^{0} \in {\scr E}\,' ({\bb R}^{n})], then [T = {\textstyle\sum_{{\bf m} \in {\bb Z}^{n}}} \tau_{\bf m} T^{0}] is a [{\bb Z}^{n}]-periodic distribution. Note that we may write [T = r * T^{0}], where [r = {\textstyle\sum_{{\bf m} \in {\bb Z}^{n}}} \delta_{({\bf m})}] consists of Dirac δ's at all nodes of the period lattice [{\bb Z}^{n}].

Conversely, any [{\bb Z}^{n}]-periodic distribution T may be written as [r * T^{0}] for some [T^{0} \in {\scr E}\,']. To retrieve such a `motif' [T^{0}] from T, a function ψ will be constructed in such a way that [\psi \in {\scr D}] (hence has compact support) and [r * \psi = 1]; then [T^{0} = \psi T]. Indicator functions (Section[link]) such as [\chi_{1}] or [\chi_{C_{1/2}}] cannot be used directly, since they are discontinuous; but regularized versions of them may be constructed by convolution (see Section[link]) as [\psi_{0} = \chi_{C_{\varepsilon}} * \theta_{\eta}], with [epsilon] and η such that [\psi_{0} ({\bf x}) = 1] on [C_{1/2}] and [\psi_{0}({\bf x}) = 0] outside [C_{3/4}]. Then the function [\psi = {\psi_{0} \over {\textstyle\sum_{{\bf m} \in {\bb Z}^{n}}} \tau_{\bf m} \psi_{0}}] has the desired property. The sum in the denominator contains at most [2^{n}] non-zero terms at any given point x and acts as a smoothly varying `multiplicity correction'.

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