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

International Tables for Crystallography (2010). Vol. B, ch. 1.3, pp. 35-36   | 1 | 2 |

Section 1.3.2.4.2.5. Convolution property

G. Bricognea

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1.3.2.4.2.5. Convolution property

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If f and g are summable, their convolution [f * g] exists and is summable, and[{\scr F}[\,f * g] ({\boldxi}) = {\textstyle\int\limits_{{\bb R}^{n}}} \left[{\textstyle\int\limits_{{\bb R}^{n}}} f({\bf y}) g({\bf x} - {\bf y}) \,\hbox{d}^{n} {\bf y}\right] \exp (-2\pi i {\boldxi} \cdot {\bf x}) \,\hbox{d}^{n} {\bf x}.]With [{\bf x} = {\bf y} + {\bf z}], so that[\exp (-2\pi i{\boldxi} \cdot {\bf x}) = \exp (-2\pi i{\boldxi} \cdot {\bf y}) \exp (-2\pi i{\boldxi} \cdot {\bf z}),]and with Fubini's theorem, rearrangement of the double integral gives:[{\scr F}[\,f * g] = {\scr F}[\,f] \times {\scr F}[g]]and similarly[\bar{\scr F}[\,f * g] = \bar{\scr F}[\,f] \times \bar{\scr F}[g].]Thus the Fourier transform and cotransform turn convolution into multiplication.








































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