Tensor product 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. 35 | 1 | 2 |

Section 1.3.2.4.2.4. Tensor product 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

1.3.2.4.2.4. 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: $[{\scr F}_{{\bf x}, \, {\bf y}} = {\scr F}_{\bf x} \otimes {\scr F}_{\bf y}.]$