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

International Tables for Crystallography (2010). Vol. B, ch. 1.3, pp. 42-43   | 1 | 2 |

## Section 1.3.2.6.4. Fourier transforms of periodic distributions

G. Bricognea

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#### 1.3.2.6.4. Fourier transforms of periodic distributions

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The content of this section is perhaps the central result in the relation between Fourier theory and crystallography (Section 1.3.4.2.1.1).

Let with r defined as in Section 1.3.2.6.2. Then , hence , so that : -periodic distributions are tempered, hence have a Fourier transform. The convolution theorem (Section 1.3.2.5.8) is applicable, giving:and similarly for .

Since , formallysay.

It is readily shown that Q is tempered and periodic, so that , while the periodicity of r implies thatSince the first factors have single isolated zeros at in , (see Section 1.3.2.3.9.4) and hence by periodicity ; convoluting with shows that . Thus we have the fundamental result: so thati.e., according to Section 1.3.2.3.9.3,

The right-hand side is a weighted lattice distribution, whose nodes are weighted by the sample values of the transform of the motif at those nodes. Since , the latter values may be writtenBy the structure theorem for distributions with compact support (Section 1.3.2.3.9.7), is a derivative of finite order of a continuous function; therefore, from Section 1.3.2.4.2.8 and Section 1.3.2.5.8, grows at most polynomially as (see also Section 1.3.2.6.10.3 about this property). Conversely, let be a weighted lattice distribution such that the weights grow at most polynomially as . Then W is a tempered distribution, whose Fourier cotransform is periodic. If T is now written as for some , then by the reciprocity theoremAlthough the choice of is not unique, and need not yield back the same motif as may have been used to build T initially, different choices of will lead to the same coefficients because of the periodicity of .

The Fourier transformation thus establishes a duality between periodic distributions and weighted lattice distributions. The pair of relationsare referred to as the Fourier analysis and the Fourier synthesis of T, respectively (there is a discrepancy between this terminology and the crystallographic one, see Section 1.3.4.2.1.1). In other words, any periodic distribution may be represented by a Fourier series (ii), whose coefficients are calculated by (i). The convergence of (ii) towards T in will be investigated later (Section 1.3.2.6.10).