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 and cotransform of a tempered distribution T are defined by for all test functions . Both and are themselves tempered distributions, since the maps and are both linear and continuous for the topology of . In the same way that x and ξ have been used consistently as arguments for ϕ and , respectively, the notation and will be used to indicate which variables are involved.

When T is a distribution with compact support, its Fourier transform may be written since the function is in while . It can be shown, as in Section 1.3.2.4.2, to be analytically continuable into an entire function over .