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

International Tables for Crystallography (2010). Vol. B, ch. 1.3, pp. 36-37   | 1 | 2 |

Section 1.3.2.4.2.10. The Paley–Wiener theorem

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

1.3.2.4.2.10. The Paley–Wiener theorem

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An extreme case of the last instance occurs when f has compact support: then [{\scr F}[\,f]] and [\bar{\scr F}[\,f]] are so regular that they may be analytically continued from [{\bb R}^{n}] to [{\bb C}^{n}] where they are entire functions, i.e. have no singularities at finite distance (Paley & Wiener, 1934[link]). This is easily seen for [{\scr F}[\,f]]: giving vector [{\boldxi} \in {\bb R}^{n}] a vector [{\boldeta} \in {\bb R}^{n}] of imaginary parts leads to[\eqalign{{\scr F}[\,f] ({\boldxi} + i{\boldeta}) &= {\textstyle\int\limits_{{\bb R}^{n}}} f({\bf x}) \exp [-2\pi i ({\boldxi} + i{\boldeta}) \cdot {\bf x}] \,\hbox{d}^{n} {\bf x}\cr &= {\scr F}[\exp (2\pi {\boldeta} \cdot {\bf x})f] ({\boldxi}),}]where the latter transform always exists since [\exp (2\pi {\boldeta} \cdot {\bf x})f] is summable with respect to x for all values of η. This analytic continuation forms the basis of the saddlepoint method in probability theory [Section 1.3.4.5.2.1[link](f)[link]] and leads to the use of maximum-entropy distributions in the statistical theory of direct phase determination [Section 1.3.4.5.2.2[link](e)[link]].

By Liouville's theorem, an entire function in [{\bb C}^{n}] cannot vanish identically on the complement of a compact subset of [{\bb R}^{n}] without vanishing everywhere: therefore [{\scr F}[\,f]] cannot have compact support if f has, and hence [{\scr D}({\bb R}^{n})] is not stable by Fourier transformation.

References

Paley, R. E. A. C. & Wiener, N. (1934). Fourier Transforms in the Complex Domain. Providence: American Mathematical Society.








































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