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

International Tables for Crystallography (2010). Vol. B, ch. 1.3, p. 39   | 1 | 2 |

Section 1.3.2.4.4.4. Symmetry property

G. Bricognea

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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[link] to extend the Fourier transformation to distributions.








































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