Symmetry property

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, p. 38 | 1 | 2 |

Section 1.3.2.4.4.4. Symmetry property

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

1.3.2.4.4.4. Symmetry property

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A final formal property of the Fourier transform, best established in $[{\scr S}]$ , is its symmetry: if f and g are in $[{\scr S}]$ , then by Fubini's theorem $[\eqalign{\langle {\scr F}[\;f], g\rangle &= {\textstyle\int\limits_{{\bb R}^{n}}} \left({\textstyle\int\limits_{{\bb R}^{n}}} f({\bf x}) \exp (-2\pi i{\boldxi} \cdot {\bf x}) \;\hbox{d}^{n}{\bf x}\right) g({\boldxi}) \;\hbox{d}^{n}{\boldxi}\cr &= {\textstyle\int\limits_{{\bb R}^{n}}} f({\bf x}) \left({\textstyle\int\limits_{{\bb R}^{n}}} g({\boldxi}) \exp (-2\pi i{\boldxi} \cdot {\bf x}) \;\hbox{d}^{n}{\boldxi}\right) \;\hbox{d}^{n}{\bf x}\cr &= \langle f, {\scr F}[g]\rangle.}]$

This possibility of `transposing' $[{\scr F}]$ (and $[\bar{\scr F}]$ ) from the left to the right of the duality bracket will be used in Section 1.3.2.5.4 to extend the Fourier transformation to distributions.

References

International Tables for Crystallography (2006). Vol. B. ch. 1.3, p. 38