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# Diffraction of Weighted Lattice Subsets

Published:2002-12-01
Printed: Dec 2002
• Michael Baake
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## Abstract

A Dirac comb of point measures in Euclidean space with bounded complex weights that is supported on a lattice $\varGamma$ inherits certain general properties from the lattice structure. In particular, its autocorrelation admits a factorization into a continuous function and the uniform lattice Dirac comb, and its diffraction measure is periodic, with the dual lattice $\varGamma^*$ as lattice of periods. This statement remains true in the setting of a locally compact Abelian group whose topology has a countable base.
 Keywords: diffraction, Dirac combs, lattice subsets, homometric sets
 MSC Classifications: 52C07 - Lattices and convex bodies in $n$ dimensions [See also 11H06, 11H31, 11P21] 43A25 - Fourier and Fourier-Stieltjes transforms on locally compact and other abelian groups 52C23 - Quasicrystals, aperiodic tilings 43A05 - Measures on groups and semigroups, etc.

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