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

© International Union of Crystallography 2006
International Tables for Crystallography (2006). Vol. B. ch. 1.3, p. 35   | 1 | 2 |

Section 1.3.2.4.2.6. Reciprocity property

G. Bricognea

a MRC Laboratory of Molecular Biology, Hills Road, Cambridge CB2 2QH, England, and LURE, Bâtiment 209D, Université Paris-Sud, 91405 Orsay, France

1.3.2.4.2.6. Reciprocity property

| top | pdf |

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'.








































to end of page
to top of page