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

International Tables for Crystallography (2006). Vol. B, ch. 1.3, pp. 46-47   | 1 | 2 |

Section 1.3.2.7.2.3. Relation between lattice distributions

G. Bricognea

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

1.3.2.7.2.3. Relation between lattice distributions

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The above relations between lattices may be rewritten in terms of the corresponding lattice distributions as follows: where and are (finite) residual-lattice distributions. We may incorporate the factor in (i) and into these distributions and define

Since , convolution with and has the effect of averaging the translates of a distribution under the elements (or cosets') of the residual lattices and , respectively. This process will be called coset averaging'. Eliminating and between (i) and (ii), and and between and , we may write: These identities show that period subdivision by convolution with (respectively ) on the one hand, and period decimation by `dilation' by on the other hand, are mutually inverse operations on and (respectively and ).