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

International Tables for Crystallography (2010). Vol. B, ch. 1.3, p. 42   | 1 | 2 |

Section 1.3.2.6.3. Identification with distributions over

G. Bricognea

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1.3.2.6.3. Identification with distributions over

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Throughout this section, periodic' will mean -periodic'.

Let , and let [s] denote the largest integer . For , let be the unique vector with . If , then if and only if . The image of the map is thus modulo , or .

If f is a periodic function over , then implies ; we may thus define a function over by putting for any such that . Conversely, if is a function over , then we may define a function f over by putting , and f will be periodic. Periodic functions over may thus be identified with functions over , and this identification preserves the notions of convergence, local summability and differentiability.

Given , we may definesince the sum only contains finitely many nonzero terms; ϕ is periodic, and . Conversely, if we may define periodic by , and by putting with ψ constructed as above.

By transposition, a distribution defines a unique periodic distribution by ; conversely, periodic defines uniquely by .

We may therefore identify -periodic distributions over with distributions over . We will, however, use mostly the former presentation, as it is more closely related to the crystallographer's perception of periodicity (see Section 1.3.4.1).