Tables for
Volume B
Reciprocal space
Edited by U. Shmueli

International Tables for Crystallography (2010). Vol. B, ch. 1.3, p. 36   | 1 | 2 |

Section Reciprocity property

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 Reciprocity property

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In general, [{\scr F}[\,f]] and [\bar{\scr F}[\,f]] are not summable, and hence cannot be further transformed; however, as they are essentially bounded, their products with the Gaussians [G_{t} (\xi) = \exp (-2\pi^{2} \|\xi\|^{2} t)] are summable for all [t \,\gt \,0], and it can be shown that[f = \lim\limits_{t\rightarrow 0} \bar{\scr F}[G_{t} {\scr F}[\,f]] = \lim\limits_{t \rightarrow 0} {\scr F}[G_{t} \bar{\scr F}[\,f]],]where the limit is taken in the topology of the [L^{1}] norm [\|.\|_{1}]. Thus [{\scr F}] and [\bar{\scr F}] are (in a sense) mutually inverse, which justifies the common practice of calling [\bar{\scr F}] the `inverse Fourier transformation'.

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