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 diffraction, Dirac combs, lattice subsets, homometric sets

MSC Classifications:

52C07, 43A25, 52C23, 43A05 show english descriptions Lattices and convex bodies in $n$ dimensions [See also 11H06, 11H31, 11P21]
Fourier and Fourier-Stieltjes transforms on locally compact and other abelian groups
Quasicrystals, aperiodic tilings
Measures on groups and semigroups, etc. 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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