Wigner transform and quasicrystals

Abstract

Quasicrystals are tempered distributions $\mu$ which satisfy symmetric conditions on $\mu$ and $\hat{\mu }$. 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 $\mu$ on ${\mathbb{R}}^d$ whose Wigner transform, $W(\mu )$, is supported on a product of two uniformly discrete sets in ${\mathbb{R}}^d$ 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 ${\mathbb{R}}^{2d}$.