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

International Tables for Crystallography (2006). Vol. B. ch. 2.1, pp. 199-200   | 1 | 2 |

Section 2.1.7.2. Mathematical background

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

2.1.7.2. Mathematical background

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Suppose that [p(x)] is a p.d.f. which accurately describes the experimental distribution of the random variable x, where x is related to a sum of random variables and can be assumed to obey (to some approximation) an ideal p.d.f., say [p^{(0)}(x)], based on the central-limit theorem. In the correction-factor approach we seek to represent [p(x)] as [p(x) = p^{(0)}(x)\textstyle\sum\limits_{k}d_{k}\; f_{k}(x), \eqno(2.1.7.1)] where [d_{k}] are coefficients which depend on the cause of the deviation of [p(x)] from the central-limit theorem approximation and [f_{k}(x)] are suitably chosen functions of x. A choice of the set [\{f_{k}\}] is deemed suitable, if only from a practical point of view, if it allows the convenient introduction of the cause of the above deviation of [p(x)] into the expansion coefficients [d_{k}]. This requirement is satisfied – also from a theoretical point of view – by taking [f_{k}(x)] as a set of polynomials which are orthogonal with respect to the ideal p.d.f., taken as their weight function (e.g. Cramér, 1951[link]). That is, the functions [f_{k}(x)] so chosen have to obey the relationship[\textstyle\int\limits_{a}^{b}f_{k}(x)f_{m}(x)p^{(0)}(x)\;{\rm d}x = \delta_{km} = \cases{ 1, & if\quad $k = m$\cr 0, & if\quad $k \neq m $}\;, \eqno(2.1.7.2)] where [[a,b]] is the range of existence of all the functions involved. It can be readily shown that the coefficients [d_{k}] are given by [d_{k} = \textstyle\int\limits_{a}^{b}f_{k}(x)p(x)\;{\rm d}x = \langle f_{k}(x) \rangle = \textstyle\sum\limits_{n = 0}^{k}c_{n}^{(k)}\langle x^{n} \rangle, \eqno(2.1.7.3)] where the brackets [\langle \; \rangle] in equation (2.1.7.3)[link] denote averaging with respect to the unknown p.d.f. [p(x)] and [c_{n}^{(k)}] is the coefficient of the nth power of x in the polynomial [f_{k}(x)]. The coefficients [d_{k}] are thus directly related to the moments of the non-ideal distribution and the coefficients of the powers of x in the orthogonal polynomials. The latter coefficients can be obtained by the Gram–Schmidt procedure (e.g. Spiegel, 1974[link]), or by direct use of the Szegö determinants (e.g. Cramér, 1951[link]), for any weight function that has finite moments. However, the feasibility of the present approach depends on our ability to obtain the moments [\langle x^{n} \rangle] without the knowledge of the non-ideal p.d.f., [p(x)].

References

First citationCramér, H. (1951). Mathematical methods of statistics. Princeton University Press.Google Scholar
First citationSpiegel, M. R. (1974). Theory and problems of Fourier analysis. Schaum's Outline Series. New York: McGraw-Hill.Google Scholar








































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