International
Tables for Crystallography Volume B Reciprocal space Edited by U. Shmueli © International Union of Crystallography 2010 
International Tables for Crystallography (2010). Vol. B, ch. 1.3, pp. 102106

The Fourier transformation plays a central role in the branch of probability theory concerned with the limiting behaviour of sums of large numbers of independent and identically distributed random variables or random vectors. This privileged role is a consequence of the convolution theorem and of the `momentgenerating' properties which follow from the exchange between differentiation and multiplication by monomials. When the limit theorems are applied to the calculation of joint probability distributions of structure factors, which are themselves closely related to the Fourier transformation, a remarkable phenomenon occurs, which leads to the saddlepoint approximation and to the maximumentropy method.
The material in this section is not intended as an introduction to probability theory [for which the reader is referred to Cramér (1946), Petrov (1975) or Bhattacharya & Rao (1976)], but only as an illustration of the role played by the Fourier transformation in certain specific areas which are used in formulating and implementing direct methods of phase determination.
The methods of probability theory just surveyed were applied to various problems formally similar to the crystallographic phase problem [e.g. the `problem of the random walk' of Pearson (1905)] by Rayleigh (1880, 1899, 1905, 1918, 1919) and Kluyver (1906). They became the basis of the statistical theory of communication with the classic papers of Rice (1944, 1945).
The Gram–Charlier and Edgeworth series were introduced into crystallography by Bertaut (1955a,b,c, 1956a) and by Klug (1958), respectively, who showed them to constitute the mathematical basis of numerous formulae derived by Hauptman & Karle (1953). The saddlepoint approximation was introduced by Bricogne (1984) and was shown to be related to variational methods involving the maximization of certain entropy criteria. This connection exhibits most of the properties of the Fourier transform at play simultaneously, and will now be described as a final illustration.
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