Multiresolution of quasicrystal diffraction spectra

Abstract: A method for analyzing and classifying two-dimensional pure point diffraction spectra (i.e. a set of Bragg peaks) of certain self-similar structures with scaling factor [beta] > 1, such as quasicrystals, is presented. The two-dimensional pure point diffraction spectrum [Pi] is viewed as a point set in the complex plane in which each point is assigned a positive number, its Bragg intensity. Then, by using a nested sequence of self-similar subsets called [beta]-lattices, we implement a multiresolution analysis of the spectrum [Pi]. This analysis yields a partition of [Pi] simultaneously in geometry, in scale and in intensity (the 'fingerprint' of the spectrum, not of the diffracting structure itself). The method is tested through numerical explorations of pure point diffraction spectra of various mathematical structures and also with the diffraction pattern of a realistic model of a quasicrystal