Diffraction spectra of weighted Delone sets on beta-lattices with beta a quadratic unitary Pisot number

Gazeau, Jean-Pierre; Verger-Gaugry, Jean-Louis

Annales de l'Institut Fourier, Tome 56 (2006) no. 7, pp. 2437-2461.

On caractérise au moyen de la théorie des distributions la transformée de Fourier d’un peigne de Dirac avec poids, plus particulièrement la partie purement ponctuelle qui correspond aux pics de Bragg dans le spectre de diffraction. La fonction intensité de ces derniers est donnée d’une manière explicite. On en déduit le spectre de diffraction d’ensembles de Delaunay avec poids supportés par les beta-réseaux dans le cas où le poids est factorisable et où beta est le nombre d’or.

The Fourier transform of a weighted Dirac comb of beta-integers is characterized within the framework of the theory of Distributions, in particular its pure point part which corresponds to the Bragg part of the diffraction spectrum. The corresponding intensity function on this Bragg part is computed. We deduce the diffraction spectrum of weighted Delone sets on beta-lattices in the split case for the weight, when beta is the golden mean.

Classification : 52C23, 78A45, 42A99
Keywords: Delone set, Meyer set, beta-integer, beta-lattice, PV number, mathematical diffraction
Mot clés : Ensembles de Delaunay, ensembles de Meyer, beta-entiers, beta-réseaux, nombres de Pisot-Vijayaraghavan, diffraction mathématique

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