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

International Tables for Crystallography (2010). Vol. B, ch. 1.3, p. 49   | 1 | 2 |

Section 1.3.2.7.2.5. Sublattice relations in terms of periodic distributions

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

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.








































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