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

International Tables for Crystallography (2010). Vol. B, ch. 1.3, p. 30   | 1 | 2 |

Section 1.3.2.3.6. Distributions associated to locally integrable functions

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

1.3.2.3.6. Distributions associated to locally integrable functions

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Let f be a complex-valued function over Ω such that [{\textstyle\int_{K}} | \,f({\bf x}) | \,\hbox{d}^{n} {\bf x}] exists for any given compact K in Ω; f is then called locally integrable.

The linear mapping from [{\scr D}(\Omega)] to [{\bb C}] defined by[\varphi \,\longmapsto\, {\textstyle\int\limits_{\Omega}} f({\bf x}) \varphi ({\bf x}) \,\hbox{d}^{n} {\bf x}]may then be shown to be continuous over [{\scr D}(\Omega)]. It thus defines a distribution [T_{f} \in {\scr D}\,'(\Omega)]:[\langle T_{f}, \varphi \rangle = {\textstyle\int\limits_{\Omega}} f({\bf x}) \varphi ({\bf x}) \,\hbox{d}^{n} {\bf x}.]As the continuity of [T_{f}] only requires that [\varphi \in {\scr D}^{(0)} (\Omega)], [T_{f}] is actually a Radon measure.

It can be shown that two locally integrable functions f and g define the same distribution, i.e.[\langle T_{f}, \varphi \rangle = \langle T_{K}, \varphi \rangle \quad \hbox{for all } \varphi \in {\scr D},]if and only if they are equal almost everywhere. The classes of locally integrable functions modulo this equivalence form a vector space denoted [L_{\rm loc}^{1} (\Omega)]; each element of [L_{\rm loc}^{1} (\Omega)] may therefore be identified with the distribution [T_{f}] defined by any one of its representatives f.








































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