Tables for
Volume B
Reciprocal space
Edited by U. Shmueli

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

Section Tensor product property

G. Bricognea

aGlobal Phasing Ltd, Sheraton House, Suites 14–16, Castle Park, Cambridge CB3 0AX, England, and LURE, Bâtiment 209D, Université Paris-Sud, 91405 Orsay, France Tensor product property

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Another elementary property of [{\scr F}] is its naturality with respect to tensor products. Let [u \in L^{1} ({\bb R}^{m})] and [v \in L^{1} ({\bb R}^{n})], and let [{\scr F}_{\bf x},{\scr F}_{\bf y},{\scr F}_{{\bf x}, \,{\bf y}}] denote the Fourier transformations in [L^{1} ({\bb R}^{m}),L^{1} ({\bb R}^{n})] and [L^{1} ({\bb R}^{m} \times {\bb R}^{n})], respectively. Then[{\scr F}_{{\bf x}, \, {\bf y}} [u \otimes v] = {\scr F}_{\bf x} [u] \otimes {\scr F}_{\bf y} [v].]Furthermore, if [f \in L^{1} ({\bb R}^{m} \times {\bb R}^{n})], then [{\scr F}_{\bf y} [\,f] \in L^{1} ({\bb R}^{m})] as a function of x and [{\scr F}_{\bf x} [\,f] \in L^{1} ({\bb R}^{n})] as a function of y, and[{\scr F}_{{\bf x}, \,{\bf y}} [\,f] = {\scr F}_{\bf x} [{\scr F}_{\bf y} [\,f]] = {\scr F}_{\bf y} [{\scr F}_{\bf x} [\,f]].]This is easily proved by using Fubini's theorem and the fact that [({\boldxi}, {\boldeta}) \cdot ({\bf x},{ \bf y}) = {\boldxi} \cdot {\bf x} + {\boldeta} \cdot {\bf y}], where [{\bf x}, {\boldxi} \in {\bb R}^{m}], [{\bf y}, {\boldeta} \in {\bb R}^{n}]. This property may be written:[{\scr F}_{{\bf x}, \, {\bf y}} = {\scr F}_{\bf x} \otimes {\scr F}_{\bf y}.]

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