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

International Tables for Crystallography (2006). Vol. B, ch. 1.3, p. 29   | 1 | 2 |

Section 1.3.2.3.2. Rationale

G. Bricognea

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

1.3.2.3.2. Rationale

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The guiding principle which leads to requiring that the functions ϕ above (traditionally called `test functions') should be well behaved is that correspondingly `wilder' behaviour can then be accommodated in the limiting behaviour of the [f_{\nu}] while still keeping the integrals [{\textstyle\int_{{\bb R}^{n}}} f_{\nu} \varphi \;\hbox{d}^{n} {\bf x}] under control. Thus

  • (i) to minimize restrictions on the limiting behaviour of the [f_{\nu}] at infinity, the ϕ's will be chosen to have compact support;

  • (ii) to minimize restrictions on the local behaviour of the [f_{\nu}], the ϕ's will be chosen infinitely differentiable.

To ensure further the continuity of functionals such as [T_{\nu}] with respect to the test function ϕ as the [f_{\nu}] go increasingly wild, very strong control will have to be exercised in the way in which a sequence [(\varphi_{j})] of test functions will be said to converge towards a limiting ϕ: conditions will have to be imposed not only on the values of the functions [\varphi_{j}], but also on those of all their derivatives. Hence, defining a strong enough topology on the space of test functions ϕ is an essential prerequisite to the development of a satisfactory theory of distributions.








































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