International
Tables for Crystallography Volume B Reciprocal space Edited by U. Shmueli © International Union of Crystallography 2006 
International Tables for Crystallography (2006). Vol. B, ch. 1.3, p. 39

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 .