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

International Tables for Crystallography (2010). Vol. B, ch. 1.3, p. 42   | 1 | 2 |

Section 1.3.2.6.3. Identification with distributions over [{\bb R}^{n}/{\bb Z}^{n}]

G. Bricognea

aGlobal Phasing Ltd, Sheraton House, Suites 14–16, Castle Park, Cambridge CB3 0AX, England, and LURE, Bâtiment 209D, Université Paris-Sud, 91405 Orsay, France

1.3.2.6.3. Identification with distributions over [{\bb R}^{n}/{\bb Z}^{n}]

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Throughout this section, `periodic' will mean `[{\bb Z}^{n}]-periodic'.

Let [s \in {\bb R}], and let [s] denote the largest integer [\leq s]. For [x = (x_{1}, \ldots, x_{n}) \in {\bb R}^{n}], let [\tilde{{\bf x}}] be the unique vector [(\tilde{x}_{1}, \ldots, \tilde{x}_{n})] with [\tilde{x}_{j} = x_{j} - [x_{j}]]. If [{\bf x},{\bf y} \in {\bb R}^{n}], then [\tilde{{\bf x}} = \tilde{{\bf y}}] if and only if [{\bf x} - {\bf y} \in {\bb Z}^{n}]. The image of the map [{\bf x} \,\longmapsto\, \tilde{{\bf x}}] is thus [{\bb R}^{n}] modulo [{\bb Z}^{n}], or [{\bb R}^{n}/{\bb Z}^{n}].

If f is a periodic function over [{\bb R}^{n}], then [\tilde{{\bf x}} = \tilde{{\bf y}}] implies [f({\bf x}) \ =] [f({\bf y})]; we may thus define a function [\tilde{f}] over [{\bb R}^{n}/{\bb Z}^{n}] by putting [\tilde{f}(\tilde{{\bf x}}) = f({\bf x})] for any [{\bf x} \in {\bb R}^{n}] such that [{\bf x} - \tilde{{\bf x}} \in {\bb Z}^{n}]. Conversely, if [\tilde{f}] is a function over [{\bb R}^{n}/{\bb Z}^{n}], then we may define a function f over [{\bb R}^{n}] by putting [f({\bf x}) = \tilde{f}(\tilde{{\bf x}})], and f will be periodic. Periodic functions over [{\bb R}^{n}] may thus be identified with functions over [{\bb R}^{n}/{\bb Z}^{n}], and this identification preserves the notions of convergence, local summability and differentiability.

Given [\varphi^{0} \in {\scr D}({\bb R}^{n})], we may define[\varphi ({\bf x}) = {\textstyle\sum\limits_{{\bf m} \in {\bb Z}^{n}}} (\tau_{\bf m} \varphi^{0}) ({\bf x})]since the sum only contains finitely many nonzero terms; ϕ is periodic, and [\tilde{\varphi} \in {\scr D}({\bb R}^{n}/{\bb Z}^{n})]. Conversely, if [\tilde{\varphi} \in {\scr D}({\bb R}^{n}/{\bb Z}^{n})] we may define [\varphi \in {\scr E}({\bb R}^{n})] periodic by [\varphi ({\bf x}) = \tilde{\varphi} (\tilde{{\bf x}})], and [\varphi^{0} \in {\scr D}({\bb R}^{n})] by putting [\varphi^{0} = \psi \varphi] with ψ constructed as above.

By transposition, a distribution [\tilde{T} \in {\scr D}\,'({\bb R}^{n}/{\bb Z}^{n})] defines a unique periodic distribution [T \in {\scr D}\,'({\bb R}^{n})] by [\langle T, \varphi^{0} \rangle = \langle \tilde{T}, \tilde{\varphi} \rangle]; conversely, [T \in {\scr D}\,'({\bb R}^{n})] periodic defines uniquely [\tilde{T} \in {\scr D}\,'({\bb R}^{n}/{\bb Z}^{n})] by [\langle \tilde{T}, \tilde{\varphi}\rangle = \langle T, \varphi^{0}\rangle].

We may therefore identify [{\bb Z}^{n}]-periodic distributions over [{\bb R}^{n}] with distributions over [{\bb R}^{n}/{\bb Z}^{n}]. 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[link]).








































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