Tables for
Volume B
Reciprocal space
Edited by U. Shmueli

International Tables for Crystallography (2010). Vol. B, ch. 1.3, p. 47   | 1 | 2 |

Section Classical [L^{2}] theory

G. Bricognea

aGlobal Phasing Ltd, Sheraton House, Suites 14–16, Castle Park, Cambridge CB3 0AX, England, and LURE, Bâtiment 209D, Université Paris-Sud, 91405 Orsay, France Classical [L^{2}] theory

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The space [L^{2}({\bb R}/{\bb Z})] of (equivalence classes of) square-integrable complex-valued functions f on the circle is contained in [L^{1}({\bb R}/{\bb Z})], since by the Cauchy–Schwarz inequality[\eqalign{\|\,f \|_{1}^{2} &= \left({\textstyle\int\limits_{0}^{1}} |\,f (x)| \times 1 \,\hbox{d}x\right)^{2} \cr &\leq \left({\textstyle\int\limits_{0}^{1}} |\,f (x)|^{2} \,\hbox{d}x\right) \left({\textstyle\int\limits_{0}^{1}} {1}^{2} \,\hbox{d}x\right) = \|\,f \|_{2}^{2} \leq \infty.}]Thus all the results derived for [L^{1}] hold for [L^{2}], a great simplification over the situation in [{\bb R}] or [{\bb R}^{n}] where neither [L^{1}] nor [L^{2}] was contained in the other.

However, more can be proved in [L^{2}], because [L^{2}] is a Hilbert space (Section[link]) for the inner product[(\,f, g) = {\textstyle\int\limits_{0}^{1}}\, \overline{f (x)} g (x) \,\hbox{d}x,]and because the family of functions [\{\exp (2 \pi imx)\}_{m \in {\bb Z}}] constitutes an orthonormal Hilbert basis for [L^{2}].

The sequence of Fourier coefficients [c_{m} (\,f)] of [f \in L^{2}] belongs to the space [\ell^{2}({\bb Z})] of square-summable sequences:[{\textstyle\sum\limits_{m \in {\bb Z}}} |c_{m} (\,f)|^{2} \,\lt\, \infty.]Conversely, every element [c = (c_{m})] of [\ell^{2}] is the sequence of Fourier coefficients of a unique function in [L^{2}]. The inner product[(c, d) = {\textstyle\sum\limits_{m \in {\bb Z}}} \overline{c_{m}} d_{m}]makes [\ell^{2}] into a Hilbert space, and the map from [L^{2}] to [\ell^{2}] established by the Fourier transformation is an isometry (Parseval/Plancherel):[\|\,f \|_{L^{2}} = \| c (\,f) \|_{{\ell}^{2}}]or equivalently:[(\,f, g) = (c (\,f), c (g)).]This is a useful property in applications, since (f, g) may be calculated either from f and g themselves, or from their Fourier coefficients [c(\,f)] and [c(g)] (see Section[link]) for crystallographic applications).

By virtue of the orthogonality of the basis [\{\exp (2 \pi imx)\}_{m \in {\bb Z}}], the partial sum [S_{p} (\,f)] is the best mean-square fit to f in the linear subspace of [L^{2}] spanned by [\{\exp (2 \pi imx)\}_{|m| \leq p}], and hence (Bessel's inequality)[{\textstyle\sum\limits_{|m| \leq p}} |c_{m} (\,f)|^{2} = \|\,f \|_{2}^{2} - {\textstyle\sum\limits_{|M| \geq p}} |c_{M} (\,f)|^{2} \leq \|\,f \|_{2}^{2}.]

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