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

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

Section 1.3.2.3.8. 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

1.3.2.3.8. Convergence of distributions

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A sequence of distributions will be said to converge in to a distribution T as if, for any given , the sequence of complex numbers converges in to the complex number .

A series of distributions will be said to converge in and to have distribution S as its sum if the sequence of partial sums converges to S.

These definitions of convergence in assume that the limits T and S are known in advance, and are distributions. This raises the question of the completeness of : if a sequence in is such that the sequence has a limit in for all , does the mapdefine a distribution ? 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 , 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 with converges to 0 in , 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 is a sequence of locally summable functions on such that

 (i) as for all ; (ii) as for all ; (iii) there exists and such that for all ν;

then the sequence of distributions converges to δ in .