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

International Tables for Crystallography (2010). Vol. B, ch. 1.3, p. 44   | 1 | 2 |

## Section 1.3.2.6.6. Duality between periodization and sampling

G. Bricognea

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#### 1.3.2.6.6. Duality between periodization and sampling

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Let be a distribution with compact support (the motif'). Its Fourier transform is analytic (Section 1.3.2.5.4) and may thus be used as a multiplier.

We may rephrase the preceding results as follows:

 (i) if is periodized by R' to give , then is sampled by ' to give ; (ii) if is sampled by ' to give , then is `periodized by R' to give .

Thus the Fourier transformation establishes a duality between the periodization of a distribution by a period lattice Λ and the sampling of its transform at the nodes of lattice reciprocal to Λ. This is a particular instance of the convolution theorem of Section 1.3.2.5.8.

At this point it is traditional to break the symmetry between and which distribution theory has enabled us to preserve even in the presence of periodicity, and to perform two distinct identifications:

 (i) a Λ-periodic distribution T will be handled as a distribution on , was done in Section 1.3.2.6.3; (ii) a weighted lattice distribution will be identified with the collection of its n-tuply indexed coefficients.