Tables for
Volume B
Reciprocal space
Edited by U. Shmueli

International Tables for Crystallography (2010). Vol. B, ch. 1.3, pp. 42-43   | 1 | 2 |

Section Fourier transforms of periodic distributions

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 Fourier transforms of periodic distributions

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The content of this section is perhaps the central result in the relation between Fourier theory and crystallography (Section[link]).

Let [T = r * T^{0}] with r defined as in Section[link]. Then [r \in {\scr S}\,'], [T^{0} \in {\scr E}\,'] hence [T^{0} \in {\scr O\,}'_{\!C}], so that [T \in {\scr S}\,']: [{\bb Z}^{n}]-periodic distributions are tempered, hence have a Fourier transform. The convolution theorem (Section[link]) is applicable, giving:[{\scr F}[T] = {\scr F}[r] \times {\scr F}[T^{0}]]and similarly for [\bar{\scr F}].

Since [{\scr F}[\delta_{({\bf m})}] (\xi) = \exp (-2 \pi i {\boldxi} \cdot {\bf m})], formally[{\scr F}[r]_{\boldxi} = {\textstyle\sum\limits_{{\bf m} \in {\bb Z}^{n}}} \exp (-2 \pi i \boldxi \cdot {\bf m}) = Q,]say.

It is readily shown that Q is tempered and periodic, so that [Q = {\textstyle\sum_{{\boldmu} \in {\bb Z}^{n}}} \tau_{{\boldmu}} (\psi Q)], while the periodicity of r implies that[[\exp (-2 \pi i \xi_{j}) - 1] \psi Q = 0, \quad j = 1, \ldots, n.]Since the first factors have single isolated zeros at [\xi_{j} = 0] in [C_{3/4}], [\psi Q = c\delta] (see Section[link]) and hence by periodicity [Q = cr]; convoluting with [\chi_{C_{1}}] shows that [c = 1]. Thus we have the fundamental result: [Scheme scheme1] so that[{\scr F}[T] = r \times {\scr F}[T^{0}]\hbox{\semi}]i.e., according to Section[link],[{\scr F}[T]_{\boldxi} = {\textstyle\sum\limits_{{\boldmu} \in {\bb Z}^{n}}} {\scr F}[T^{0}] ({\boldmu}) \times \delta_{({\boldmu})}.]

The right-hand side is a weighted lattice distribution, whose nodes [{\boldmu} \in {\bb Z}^{n}] are weighted by the sample values [{\scr F}[T^{0}] ({\boldmu})] of the transform of the motif [T^{0}] at those nodes. Since [T^{0} \in {\scr E}\,'], the latter values may be written[{\scr F}[T^{0}]({\boldmu}) = \langle T_{\bf x}^{0}, \exp (-2 \pi i {\boldmu} \cdot {\bf x})\rangle.]By the structure theorem for distributions with compact support (Section[link]), [T^{0}] is a derivative of finite order of a continuous function; therefore, from Section[link] and Section[link], [{\scr F}[T^{0}]({\boldmu})] grows at most polynomially as [\|{\boldmu}\| \rightarrow \infty] (see also Section[link] about this property). Conversely, let [W = {\textstyle\sum_{{\boldmu} \in {\bb Z}^{n}}} w_{{\boldmu}} \delta_{({\boldmu})}] be a weighted lattice distribution such that the weights [w_{\boldmu}] grow at most polynomially as [\|{\boldmu}\| \rightarrow \infty]. Then W is a tempered distribution, whose Fourier cotransform [T_{\bf x} = {\textstyle\sum_{{\boldmu} \in {\bb Z}^{n}}} w_{\boldmu} \exp (+2 \pi i {\boldmu} \cdot {\bf x})] is periodic. If T is now written as [r * T^{0}] for some [T^{0} \in {\scr E}\,'], then by the reciprocity theorem[w_{\boldmu} = {\scr F}[T^{0}]({\boldmu}) = \langle T_{\bf x}^{0}, \exp (-2 \pi i {\boldmu} \cdot {\bf x})\rangle.]Although the choice of [T^{0}] is not unique, and need not yield back the same motif as may have been used to build T initially, different choices of [T^{0}] will lead to the same coefficients [w_{\boldmu}] because of the periodicity of [\exp (-2 \pi i {\boldmu} \cdot {\bf x})].

The Fourier transformation thus establishes a duality between periodic distributions and weighted lattice distributions. The pair of relations[\displaylines{\quad (\hbox{i})\hfill w_{\boldmu} = \langle T_{\bf x}^{0}, \exp (-2 \pi i {\boldmu} \cdot {\bf x})\rangle \quad\hfill\cr \quad(\hbox{ii})\hfill T_{\bf x} = {\textstyle\sum\limits_{{\boldmu} \in {\bb Z}^{n}}} w_{\boldmu} \exp (+2 \pi i {\boldmu} \cdot {\bf x}) \hfill}]are referred to as the Fourier analysis and the Fourier synthesis of T, respectively (there is a discrepancy between this terminology and the crystallographic one, see Section[link]). In other words, any periodic distribution [T \in {\scr S}\,'] may be represented by a Fourier series (ii), whose coefficients are calculated by (i). The convergence of (ii) towards T in [{\scr S}\,'] will be investigated later (Section[link]).

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