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

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#### 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 and are so regular that they may be analytically continued from to where they are entire functions, i.e. have no singularities at finite distance (Paley & Wiener, 1934). This is easily seen for : giving vector a vector of imaginary parts leads towhere the latter transform always exists since 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(f)] and leads to the use of maximum-entropy distributions in the statistical theory of direct phase determination [Section 1.3.4.5.2.2(e)].

By Liouville's theorem, an entire function in cannot vanish identically on the complement of a compact subset of without vanishing everywhere: therefore cannot have compact support if f has, and hence 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.