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

International Tables for Crystallography (2006). Vol. B, ch. 1.3, p. 39   | 1 | 2 |

Section 1.3.2.5.4. Fourier transforms of tempered distributions

G. Bricognea

aMRC Laboratory of Molecular Biology, Hills Road, Cambridge CB2 2QH, England, and LURE, Bâtiment 209D, Université Paris-Sud, 91405 Orsay, France

1.3.2.5.4. Fourier transforms of tempered distributions

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The Fourier transform [{\scr F}[T]] and cotransform [\bar{\scr F}[T]] of a tempered distribution T are defined by [\eqalign{\langle {\scr F}[T], \varphi \rangle &= \langle T, {\scr F}[\varphi]\rangle \cr \langle \bar{\scr F}[T], \varphi \rangle &= \langle T, \bar{\scr F}[\varphi]\rangle}] for all test functions [\varphi \in {\scr S}]. Both [{\scr F}[T]] and [\bar{\scr F}[T]] are themselves tempered distributions, since the maps [\varphi \;\longmapsto\; {\scr F}[\varphi]] and [\varphi \;\longmapsto\; \bar{\scr F}[\varphi]] are both linear and continuous for the topology of [{\scr S}]. In the same way that x and ξ have been used consistently as arguments for ϕ and [{\scr F}[\varphi]], respectively, the notation [T_{\bf x}] and [{\scr F}[T]_{\boldxi}] will be used to indicate which variables are involved.

When T is a distribution with compact support, its Fourier transform may be written [{\scr F}[T_{\bf x}]_{\boldxi} = \langle T_{\bf x}, \exp (- 2\pi i \boldxi \cdot {\bf x})\rangle] since the function [{\bf x} \;\longmapsto\; \exp (- 2\pi i {\boldxi} \cdot {\bf x})] is in [{\scr E}] while [T_{\bf x} \in {\scr E}\,']. It can be shown, as in Section 1.3.2.4.2[link], to be analytically continuable into an entire function over [{\bb C}^{n}].








































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