Tables for
Volume B
Reciprocal space
Edited by U. Shmueli

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

Section Duality between periodization and sampling

G. Bricognea

aGlobal Phasing Ltd, Sheraton House, Suites 14–16, Castle Park, Cambridge CB3 0AX, England, and LURE, Bâtiment 209D, Université Paris-Sud, 91405 Orsay, France Duality between periodization and sampling

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Let [T^{0}] be a distribution with compact support (the `motif'). Its Fourier transform [\bar{\scr F}[T^{0}]] is analytic (Section[link]) and may thus be used as a multiplier.

We may rephrase the preceding results as follows:

  • (i) if [T^{0}] is `periodized by R' to give [R * T^{0}], then [\bar{\scr F}[T^{0}]] is `sampled by [R^{*}]' to give [|\!\det {\bf A}|^{-1} R^{*} \cdot \bar{\scr F}[T^{0}]];

  • (ii) if [\bar{\scr F}[T^{0}]] is `sampled by [R^{*}]' to give [R^{*} \cdot \bar{\scr F}[T^{0}]], then [T^{0}] is `periodized by R' to give [|\!\det {\bf A}| R * T^{0}].

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 [\Lambda^{*}] reciprocal to Λ. This is a particular instance of the convolution theorem of Section[link].

At this point it is traditional to break the symmetry between [{\scr F}] and [\bar{\scr F}] 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 [\tilde{T}] on [{\bb R}^{n} / \Lambda], was done in Section[link];

  • (ii) a weighted lattice distribution [W = {\textstyle\sum_{{\boldmu} \in {\bb Z}^{n}}} W_{\boldmu} \delta_{[({\bf A}^{-1})^{T} {\boldmu}]}] will be identified with the collection [\{W_{\boldmu}|{\boldmu} \in {\bb Z}^{n}\}] of its n-tuply indexed coefficients.

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