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 1.3.2.6.2. -periodic distributions in

G. Bricognea

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#### 1.3.2.6.2. -periodic distributions in

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A distribution is called periodic with period lattice (or -periodic) if for all (in crystallography the period lattice is the direct lattice).

Given a distribution with compact support , then is a -periodic distribution. Note that we may write , where consists of Dirac δ's at all nodes of the period lattice .

Conversely, any -periodic distribution T may be written as for some . To retrieve such a motif' from T, a function ψ will be constructed in such a way that (hence has compact support) and ; then . Indicator functions (Section 1.3.2.2) such as or cannot be used directly, since they are discontinuous; but regularized versions of them may be constructed by convolution (see Section 1.3.2.3.9.7) as , with and η such that on and outside . Then the functionhas the desired property. The sum in the denominator contains at most nonzero terms at any given point x and acts as a smoothly varying multiplicity correction'.