Mathematical background

Shmueli, U.; Wilson, A. J. C.

doi:10.1107/97809553602060000554

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

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

Section 2.1.7.2. Mathematical background

U. Shmueli^a ^* and A. J. C. Wilson^b ^‡

^a School 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). 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) 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), or by direct use of the Szegö determinants (e.g. Cramér, 1951), 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)] .