Wigner transform and quasicrystals

Fourier quasicrystals are tempered distributions μ which satisfy symmetric conditions on μ and μˆ. This suggests that techniques from time-frequency analysis could possibly be useful tools in the study of such structures. In this paper we explore this direction considering quasicrystals type conditions on time-frequency representations instead of separately on the distribution and its Fourier transform. More precisely we prove that a tempered distribution μ on Rd whose Wigner transform, W(μ), is supported on a product of two uniformly discrete sets in Rd is a quasicrystal. This result is partially extended to a generalization of the Wigner transform, called matrix-Wigner transform which is defined in terms of the Wigner transform and a linear map T on R2.