Duality between periodization and sampling

Bricogne, G.

doi:10.1107/97809553602060000551

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

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International Tables for Crystallography (2006). Vol. B. ch. 1.3, p. 42 | 1 | 2 |

Section 1.3.2.6.6. Duality between periodization and sampling

G. Bricogne^a

^a MRC Laboratory of Molecular Biology, Hills Road, Cambridge CB2 2QH, England, and LURE, Bâtiment 209D, Université Paris-Sud, 91405 Orsay, France

1.3.2.6.6. 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 1.3.2.5.4) 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 1.3.2.5.8.

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 1.3.2.6.3;
(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.

References

International Tables for Crystallography (2006). Vol. B. ch. 1.3, p. 42