Time-frequency analysis and Fourier quasicrystals

Fourier analysis is a fundamental mathematical tool for the study of ordered periodic structures such as crystals, as well as for detecting the aperiodicity exhibited by quasicrystals. In this framework, spatial structure and diffraction properties are described in terms of discrete measures and their Fourier transforms. Motivated by the ability of time–frequency representations to provide simultaneous localization of acoustic signals both with respect to time and frequency, we propose their use in the analysis of quasicrystals. In particular, after a review of the main time-frequency representations, we show how some of these quadratic forms, especially Wigner-type transforms, offer effective tools for detecting and characterizing Fourier quasicrystals. Several recent results are reviewed and discussed within a unified time-frequency framework.