Decrease at infinity

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. 36 | 1 | 2 |

Section 1.3.2.4.2.9. Decrease at infinity

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

1.3.2.4.2.9. Decrease at infinity

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Conversely, assume that f is summable on $[{\bb R}^{n}]$ and that f decreases fast enough at infinity for $[{\bf x}^{{\bf m}} f]$ also to be summable, for some multi-index m. Then the integral defining $[{\scr F}[\;f]]$ may be subjected to the differential operator $[D^{{\bf m}}]$ , still yielding a convergent integral: therefore $[D^{{\bf m}} {\scr F}[\;f]]$ exists, and $[D^{{\bf m}} ({\scr F}[\;f]) ({\boldxi}) = {\scr F}[(-2\pi i{\bf x})^{{\bf m}} f] ({\boldxi})]$ with the bound $[\|D^{{\bf m}} ({\scr F}[\;f])\|_{\infty} = \|(2\pi {\bf x})^{{\bf m}} f\|_{1}.]$

Similar results hold for $[\bar{\scr F}]$ , with $[-2\pi i {\bf x}]$ replaced by $[2\pi i{\bf x}]$ . Thus, the faster f decreases at infinity, the more $[{\scr F}[\;f]]$ and $[\bar{\scr F}[\;f]]$ are differentiable, with bounded derivatives. This property is the converse of that described in Section 1.3.2.4.2.8, and their combination is fundamental in the definition of the function space $[{\scr S}]$ in Section 1.3.2.4.4.1, of tempered distributions in Section 1.3.2.5, and in the extension of the Fourier transformation to them.

References

International Tables for Crystallography (2006). Vol. B. ch. 1.3, p. 36