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Wigner transform and quasicrystals
- Publication Year :
- 2021
-
Abstract
- Quasicrystals are tempered distributions $\mu$ which satisfy symmetric conditions on $\mu$ and $\widehat \mu$. 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 ${\mathbb R}^{2d}$.
- Subjects :
- Mathematics - Functional Analysis
Subjects
Details
- Database :
- arXiv
- Publication Type :
- Report
- Accession number :
- edsarx.2106.09364
- Document Type :
- Working Paper