Tables for
Volume B
Reciprocal space
Edited by U. Shmueli

International Tables for Crystallography (2010). Vol. B, ch. 1.3, p. 40   | 1 | 2 |

Section Transposition of basic properties

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 Transposition of basic properties

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The duality between differentiation and multiplication by a monomial extends from [{\scr S}] to [{\scr S}\,'] by transposition:[\eqalign{{\scr F}[D_{\bf x}^{{\bf p}} T_{\bf x}]_{\boldxi} &= (2\pi i \boldxi)^{{\bf p}} {\scr F}[T_{\bf x}]_{\boldxi} \cr D_{\boldxi}^{{\bf p}} ({\scr F}[T_{\bf x}]_{\boldxi}) &= {\scr F}[(- 2\pi i {\bf x})^{{\bf p}} T_{\bf x}]_{\boldxi}.}]Analogous formulae hold for [\bar{\scr F}], with i replaced by −i.

The formulae expressing the duality between translation and phase shift, e.g.[\eqalign{{\scr F}[\tau_{\bf a} T_{\bf x}]_{\boldxi} &= \exp (-2\pi i{\bf a} \cdot {\boldxi}) {\scr F}[T_{\bf x}]_{\boldxi} \cr \tau_{\boldalpha} ({\scr F}[T_{\bf x}]_{\boldxi}) &= {\scr F}[\exp (2\pi i{\boldalpha} \cdot {\bf x}) T_{\bf x}]_{\boldxi}\hbox{\semi}}]between a linear change of variable and its contragredient, e.g.[{\scr F}[A^{\#} T] = |\hbox{det } {\bf A}| [({\bf A}^{-1})^{T}]^{\#} {\scr F}[T]\hbox{\semi}]are obtained similarly by transposition from the corresponding identities in [{\scr S}]. They give a transposition formula for an affine change of variables [{\bf x} \,\longmapsto\, S({\bf x}) = {\bf Ax} + {\bf b}] with nonsingular matrix A:[\eqalign{{\scr F}[S^{\#} T] &= \exp (-2\pi i{\boldxi} \cdot {\bf b}) {\scr F}[A^{\#} T] \cr &= \exp (-2\pi i{\boldxi} \cdot {\bf b}) |\hbox{det } {\bf A}| [({\bf A}^{-1})^{T}]^{\#} {\scr F}[T],}]with a similar result for [\bar{\scr F}], replacing −i by +i.

Conjugate symmetry is obtained similarly:[{\scr F}[\bar{T}] = \breve{\overline{{\scr F}[T]}}, {\scr F}[\breve{\bar{T}}] = \overline{{\scr F}[T]},]with the same identities for [\bar{\scr F}].

The tensor product property also transposes to tempered distributions: if [U \in {\scr S}\,'({\bb R}^{m}), V \in {\scr S}\,'({\bb R}^{n})],[\eqalign{{\scr F}[U_{\bf x} \otimes V_{\bf y}] &= {\scr F}[U]_{\boldxi} \otimes {\scr F}[V]_{\boldeta} \cr \bar{\scr F}[U_{\bf x} \otimes V_{\bf y}] &= \bar{\scr F}[U]_{\boldxi} \otimes \bar{\scr F}[V]_{\boldeta}.}]

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