Fourier transforms of periodic distributions

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, pp. 41-42 | 1 | 2 |

Section 1.3.2.6.4. Fourier transforms of periodic distributions

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

1.3.2.6.4. 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 1.3.4.2.1.1).

Let $[T = r * T^{0}]$ with r defined as in Section 1.3.2.6.2. 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 1.3.2.5.8) 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 1.3.2.3.9.4) 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 $[{\scr F}[T] = r \times {\scr F}[T^{0}]\hbox{;}]$ i.e., according to Section 1.3.2.3.9.3, $[{\scr F}[T]_{\boldxi} = {\textstyle\sum\limits_{{\boldmu} \in {\bb Z}^{n}}} {\scr F}[T^{0}] ({\boldmu}) \times \delta_{({\boldmu})}.]$

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 1.3.2.3.9.7), $[T^{0}]$ is a derivative of finite order of a continuous function; therefore, from Section 1.3.2.4.2.8 and Section 1.3.2.5.8, $[{\scr F}[T^{0}]({\boldmu})]$ grows at most polynomially as $[\|{\boldmu}\| \rightarrow \infty]$ (see also Section 1.3.2.6.10.3 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 1.3.4.2.1.1). 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 1.3.2.6.10).

References

International Tables for Crystallography (2006). Vol. B. ch. 1.3, pp. 41-42