Tables for
Volume B
Reciprocal space
Edited by U. Shmueli

International Tables for Crystallography (2006). Vol. B, ch. 1.3, p. 30   | 1 | 2 |

Section Definition of distributions

G. Bricognea

aMRC Laboratory of Molecular Biology, Hills Road, Cambridge CB2 2QH, England, and LURE, Bâtiment 209D, Université Paris-Sud, 91405 Orsay, France Definition of distributions

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A distribution T on Ω is a linear form over [{\scr D}(\Omega)], i.e. a map [T: \varphi \;\longmapsto\; \langle T, \varphi \rangle] which associates linearly a complex number [\langle T, \varphi \rangle] to any [\varphi \in {\scr D}(\Omega)], and which is continuous for the topology of that space. In the terminology of Section[link], T is an element of [{\scr D}\,'(\Omega)], the topological dual of [{\scr D}(\Omega)].

Continuity over [{\scr D}] is equivalent to continuity over [{\scr D}_{K}] for all compact K contained in Ω, and hence to the condition that for any sequence [(\varphi_{\nu})] in [{\scr D}] such that

  • (i) Supp [\varphi_{\nu}] is contained in some compact K independent of ν,

  • (ii) the sequences [(|D^{\bf p} \varphi_{\nu}|)] converge uniformly to 0 on K for all multi-indices p;

then the sequence of complex numbers [\langle T, \varphi_{\nu}\rangle] converges to 0 in [{\bb C}].

If the continuity of a distribution T requires (ii)[link] for [|{\bf p}| \leq m] only, T may be defined over [{\scr D}^{(m)}] and thus [T \in {\scr D}\,'^{(m)}]; T is said to be a distribution of finite order m. In particular, for [m = 0, {\scr D}^{(0)}] is the space of continuous functions with compact support, and a distribution [T \in {\scr D}\,'^{(0)}] is a (Radon) measure as used in the theory of integration. Thus measures are particular cases of distributions.

Generally speaking, the larger a space of test functions, the smaller its topological dual: [m \;\lt\; n \Rightarrow {\scr D}^{(m)} \supset {\scr D}^{(n)} \Rightarrow {\scr D}\,'^{(n)} \supset {\scr D}\,'^{(m)}.] This clearly results from the observation that if the ϕ's are allowed to be less regular, then less wildness can be accommodated in T if the continuity of the map [\varphi \;\longmapsto\; \langle T, \varphi \rangle] with respect to ϕ is to be preserved.

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