Sublattice relations in terms of periodic distributions

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. 47 | 1 | 2 |

Section 1.3.2.7.2.5. Sublattice relations in terms of periodic distributions

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

1.3.2.7.2.5. Sublattice relations in terms of periodic distributions

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The usual presentation of this duality is not in terms of lattice distributions, but of periodic distributions obtained by convolving them with a motif.

Given $[T^{0} \in {\scr E}\,' ({\bb R}^{n})]$ , let us form $[R_{\bf A} * T^{0}]$ , then decimate its transform $[(1/|\det {\bf A}|) R_{\bf A}^{*} \times \bar{\scr F}[T^{0}]]$ by keeping only its values at the points of the coarser lattice $[\Lambda_{\bf B}^{*} = {\bf D}^{T} \Lambda_{\bf A}^{*}]$ ; as a result, $[R_{\bf A}^{*}]$ is replaced by $[(1/|\det {\bf D}|) R_{\bf B}^{*}]$ , and the reverse transform then yields $[\displaylines{\hfill{1 \over |\det {\bf D}|} R_{\bf B} * T^{0} = S_{{\bf B}/{\bf A}} * (R_{\bf A} * T^{0})\hfill \hbox{by (ii)},}]$ which is the coset-averaged version of the original $[R_{\bf A} * T^{0}]$ . The converse situation is analogous to that of Shannon's sampling theorem. Let a function $[\varphi \in {\scr E}({\bb R}^{n})]$ whose transform $[\Phi = {\scr F}[\varphi]]$ has compact support be sampled as $[R_{\bf B} \times \varphi]$ at the nodes of $[\Lambda_{\bf B}]$ . Then $[{\scr F}[R_{\bf B} \times \varphi] = {1 \over |\det {\bf B}|} (R_{\bf B}^{*} * \Phi)]$ is periodic with period lattice $[\Lambda_{\bf B}^{*}]$ . If the sampling lattice $[\Lambda_{\bf B}]$ is decimated to $[\Lambda_{\bf A} = {\bf D} \Lambda_{\bf B}]$ , the inverse transform becomes $[\eqalign{{\hbox to 48pt{}}{\scr F}[R_{\bf A} \times \varphi] &= {1 \over |\det {\bf D}|} (R_{\bf A}^{*} * \Phi)\cr &= S_{{\bf A}/{\bf B}}^{*} * (R_{\bf B}^{*} * \Phi){\hbox to 58pt{}}\hbox{by (ii)}^{*},}]$ hence becomes periodized more finely by averaging over the cosets of $[\Lambda_{\bf A}^{*}/\Lambda_{\bf B}^{*}]$ . With this finer periodization, the various copies of Supp Φ may start to overlap (a phenomenon called `aliasing'), indicating that decimation has produced too coarse a sampling of φ.