-periodic distributions in

Bricogne, G.

doi:10.1107/97809553602060000551

International
Tables for
Crystallography
Volume B
Reciprocal space
Edited by U. Shmueli

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

Section 1.3.2.6.2. $[{\bb Z}^{n}]$ -periodic distributions in $[{\bb R}^{n}]$

G. Bricogne^a

^a MRC Laboratory of Molecular Biology, Hills Road, Cambridge CB2 2QH, England, and LURE, Bâtiment 209D, Université Paris-Sud, 91405 Orsay, France

1.3.2.6.2. $[{\bb Z}^{n}]$ -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 1.3.2.2) 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 1.3.2.3.9.7) as $[\psi_{0} = \chi_{C_{\varepsilon}} * \theta_{\eta}]$ , with ɛ 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'.