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

International Tables for Crystallography (2010). Vol. B, ch. 1.3, p. 29   | 1 | 2 |

## Section 1.3.2.3.2. Rationale

G. Bricognea

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#### 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 while still keeping the integrals under control. Thus

 (i) to minimize restrictions on the limiting behaviour of the at infinity, the ϕ's will be chosen to have compact support; (ii) to minimize restrictions on the local behaviour of the , the ϕ's will be chosen infinitely differentiable.

To ensure further the continuity of functionals such as with respect to the test function ϕ as the go increasingly wild, very strong control will have to be exercised in the way in which a sequence 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 , 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.