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. 38   | 1 | 2 |

Section 1.3.2.4.5. Various writings of Fourier transforms

G. Bricognea

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

1.3.2.4.5. Various writings of Fourier transforms

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Other ways of writing Fourier transforms in [{\bb R}^{n}] exist besides the one used here. All have the form [{\scr F}_{h, \, \omega}[\;f]({\boldxi}) = {1 \over h^{n}} {\int\limits_{{\bb R}^{n}}} f({\bf x}) \exp (-i\omega {\boldxi} \cdot {\bf x}) \;\hbox{d}^{n}{\bf x},] where h is real positive and ω real non-zero, with the reciprocity formula written: [f({\bf x}) = {1 \over k^{n}} {\int\limits_{{\bb R}^{n}}} {\scr F}_{h, \,\omega}[\;f]({\boldxi}) \exp (+i\omega {\boldxi} \cdot {\bf x}) \;\hbox{d}^{n}{\bf x}] with k real positive. The consistency condition between h, k and ω is [hk = {2\pi \over |\omega|}.]

The usual choices are: [\displaylines{\quad (\hbox{i})\quad\; \omega = \pm 2 \pi, h = k = 1 {\hbox to 18pt{}} (\hbox{as here})\hbox{;}\hfill\cr \quad (\hbox{ii})\quad\omega = \pm 1, h = 1, k = 2 \pi {\hbox to 9.5pt{}} (\hbox{in probability theory}\hfill\cr \quad \phantom{(\hbox{ii})\quad\omega = \pm 1, h = 1, k = 2 \pi {\hbox to 10pt{}}} \hbox{and in solid-state physics})\hbox{;}\hfill\cr \quad(\hbox{iii})\;\;\; \omega = \pm 1, h = k = \sqrt{2 \pi} {\hbox to 10pt{}} (\hbox{in much of classical analysis}).\hfill}]

It should be noted that conventions (ii) and (iii) introduce numerical factors of 2π in convolution and Parseval formulae, while (ii) breaks the symmetry between [{\scr F}] and [\bar{\scr F}].








































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