Identification with distributions over

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

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

1.3.2.6.3. 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 non-zero 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).

References

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