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

International Tables for Crystallography (2010). Vol. B, ch. 1.3, p. 41   | 1 | 2 |

Section 1.3.2.5.7. Reciprocity theorem

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

1.3.2.5.7. Reciprocity theorem

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The previous results now allow a self-contained and rigorous proof of the reciprocity theorem between [{\scr F}] and [\bar{\scr F}] to be given, whereas in traditional settings (i.e. in [L^{1}] and [L^{2}]) the implicit handling of δ through a limiting process is always the sticking point.

Reciprocity is first established in [{\scr S}] as follows:[\eqalign{\bar{\scr F}[{\scr F}[\varphi]] ({\bf x}) &= {\textstyle\int\limits_{{\bb R}^{n}}} {\scr F}[\varphi] ({\boldxi}) \exp (2\pi i{\boldxi} \cdot {\bf x})\ {\rm d}^{n} {\boldxi} \cr &= {\textstyle\int\limits_{{\bb R}^{n}}} {\scr F}[\tau_{-{\bf x}} \varphi] ({\boldxi})\ {\rm d}^{n} {\boldxi} \cr &= \langle 1, {\scr F}[\tau_{-{\bf x}} \varphi]\rangle \cr &= \langle {\scr F}[1], \tau_{-{\bf x}} \varphi\rangle \cr &= \langle \tau_{\bf x} \delta, \varphi\rangle \cr &= \varphi ({\bf x})}]and similarly[{\scr F}[\bar{\scr F}[\varphi]] ({\bf x}) = \varphi ({\bf x}).]

The reciprocity theorem is then proved in [{\scr S}\,'] by transposition:[\bar{\scr F}[{\scr F}[T]] = {\scr F}[\bar{\scr F}[T]] = T \quad\hbox{for all } T \in {\scr S}\,'.]Thus the Fourier cotransformation [\bar{\scr F}] in [{\scr S}\,'] may legitimately be called the `inverse Fourier transformation'.

The method of Section 1.3.2.4.3[link] may then be used to show that [{\scr F}] and [\bar{\scr F}] both have period 4 in [{\scr S}\,'].








































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