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

International Tables for Crystallography (2006). Vol. B. ch. 2.1, p. 203   | 1 | 2 |

Section 2.1.7.4. Fourier versus Hermite approximations

U. Shmuelia* and A. J. C. Wilsonb

aSchool of Chemistry, Tel Aviv University, Tel Aviv 69 978, Israel, and bSt John's College, Cambridge, England
Correspondence e-mail:  ushmueli@post.tau.ac.il

2.1.7.4. Fourier versus Hermite approximations

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As noted in Section 2.1.8.7[link] below, the Fourier representation of the probability distribution of [|F|] is usually much better than the particular orthogonal-function representation discussed in Section 2.1.7.3[link]. Many, perhaps most, non-ideal centric distributions look like slight distortions of the ideal (Gaussian) distribution and have no resemblance to a cosine function. The empirical observation thus seems paradoxical. The probable explanation has been pointed out by Wilson (1986b[link]). A truncated Fourier series is a best approximation, in the least-squares sense, to the function represented. The particular orthogonal-function approach used in equation (2.1.7.5)[link], on the other hand, is not a least-squares approximation to [p_{c}(|E|)], but is a least-squares approximation to [p_{c}(|E|)\exp(|E|^{2}/4). \eqno(2.1.7.15)] The usual expansions (often known as Gram–Charlier or Edgeworth) thus give great weight to fitting the distribution of the (compararively few) strong reflections, at the expense of a poor fit for the (much more numerous) weak-to-medium ones. Presumably, a similar situation exists for the representation of acentric distributions, but this has not been investigated in detail. Since the centric distributions [p_{c}(|E|)] often look nearly Gaussian, one is led to ask if there is an expansion in orthogonal functions that (i) has the leading term [p_{c}(|E|)] and (ii) is a least-squares (as well as an orthogonal-function)2 fit to [p_{c}(|E|)]. One does exist, based on the orthogonal functions [f_{k} = n(x)He_{k}(2^{1/2}x), \eqno(2.1.7.16)] where [n(x)] is the Gaussian distribution (Myller-Lebedeff, 1907[link]). Unfortunately, no reasonably simple relationship between the coefficients [d_{k}] and readily evaluated properties of [p_{c}(|E|)] has been found, and the Myller-Lebedeff expansion has not, as yet, been applied in crystallography. Although Stuart & Ord (1994[link], p. 112) dismiss it in a three-line footnote, it does have important applications in astronomy (van der Marel & Franx, 1993[link]; Gerhard, 1993[link]).

References

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First citationMarel, R. P. van der & Franx, M. (1993). A new method for the identification of non-Gaussian line profiles in elliptical galaxies. Astrophys. J. 407, 525–539.Google Scholar
First citationMyller-Lebedeff, W. (1907). Die Theorie der Integralgleichungen in Anwendung auf einige Reihenentwicklungen. Math. Ann. 64, 388–416.Google Scholar
First citationStuart, A. & Ord, K. (1994). Kendall's advanced theory of statistics. Vol. 1. Distribution theory, 6th ed. London: Edward Arnold.Google Scholar
First citationWilson, A. J. C. (1986b). Fourier versus Hermite representations of probability distributions. Acta Cryst. A42, 81–83.Google Scholar








































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