Tables for
Volume B
Reciprocal space
Edited by U. Shmueli

International Tables for Crystallography (2010). Vol. B, ch. 1.3, pp. 30-31   | 1 | 2 |

Section Convergence of distributions

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 Convergence of distributions

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A sequence [(T_{j})] of distributions will be said to converge in [{\scr D}\,'] to a distribution T as [j \rightarrow \infty] if, for any given [\varphi \in {\scr D}], the sequence of complex numbers [(\langle T_{j}, \varphi \rangle)] converges in [{\bb C}] to the complex number [\langle T, \varphi \rangle].

A series [{\textstyle\sum_{j=0}^{\infty}} T_{j}] of distributions will be said to converge in [{\scr D}\,'] and to have distribution S as its sum if the sequence of partial sums [S_{k} = {\textstyle\sum_{j=0}^{k}}] converges to S.

These definitions of convergence in [{\scr D}\,'] assume that the limits T and S are known in advance, and are distributions. This raises the question of the completeness of [{\scr D}\,']: if a sequence [(T_{j})] in [{\scr D}\,'] is such that the sequence [(\langle T_{j}, \varphi \rangle)] has a limit in [{\bb C}] for all [\varphi \in {\scr D}], does the map[\varphi \,\longmapsto\, \lim_{j \rightarrow \infty} \langle T_{j}, \varphi \rangle]define a distribution [T \in {\scr D}\,']? In other words, does the limiting process preserve continuity with respect to ϕ? It is a remarkable theorem that, because of the strong topology on [{\scr D}], this is actually the case. An analogous statement holds for series. This notion of convergence does not coincide with any of the classical notions used for ordinary functions: for example, the sequence [(\varphi_{\nu})] with [\varphi_{\nu} (x) = \cos \nu x] converges to 0 in [{\scr D}\,'({\bb R})], but fails to do so by any of the standard criteria.

An example of convergent sequences of distributions is provided by sequences which converge to δ. If [(\,f_{\nu})] is a sequence of locally summable functions on [{\bb R}^{n}] such that

  • (i) [\textstyle{\int_{\|{\bf x}\| \lt\, b}} \,f_{\nu} ({\bf x}) \,\hbox{d}^{n} {\bf x} \rightarrow 1] as [\nu \rightarrow \infty] for all [b \,\gt \,0];

  • (ii) [{\textstyle\int_{a \leq \|{\bf x}\| \leq 1/a}} |\,f_{\nu} ({\bf x})| \,\hbox{d}^{n} {\bf x} \rightarrow 0] as [\nu \rightarrow \infty] for all [0 \,\lt\, a \,\lt\, 1];

  • (iii) there exists [d \,\gt\, 0] and [M \,\gt \,0] such that [{\textstyle\int_{\|{\bf x}\|\lt\, d}} |\,f_{\nu} ({\bf x})| \,\hbox{d}^{n} {\bf x}\,\lt \,M] for all ν;

then the sequence [(T_{f_{\nu}})] of distributions converges to δ in [{\scr D}\,'({\bb R}^{n})].

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