PreprintCC0arXiv
Some properties of Fourier quasicrystals and measures on a strip
Abstract
We extend certain results of the theory of Fourier quasicrystals on the real line to the case of a horizontal strip of finite width. For measures in a strip we use a natural generalization of the usual Fourier transform for measures on the line. We consider positive or translation bounded measures μ on a strip whose Fourier transform is a pure point measure μ̂=∑γ∈Γbγδγ (as usual, δγ is the unit mass at the point γ). We prove that the measure ν=∑γ∈Γ|bγ|2δγ has the exponential growth. Moreover, if for some η>0 the points of Γ in every interval of length η are linearly independent over integers, then the measure μ̂ also has the exponential growth.
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BibTeX
@misc{favorov2026some,
title = {{Some properties of Fourier quasicrystals and measures on a strip}},
author = {Sergii Favorov and Özkan Deǧer},
howpublished = {arXiv},
year = {2026},
month = sep,
doi = {10.48550/arxiv.2605.10766},
eprint = {2605.10766},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2605.10766},
note = {CC0},
}
RIS
TY - UNPB
TI - Some properties of Fourier quasicrystals and measures on a strip
AU - Sergii Favorov
AU - Özkan Deǧer
PY - 2026
DA - 2026/09/18
PB - arXiv
DO - 10.48550/arxiv.2605.10766
UR - https://arxiv.org/abs/2605.10766
AB - We extend certain results of the theory of Fourier quasicrystals on the real line to the case of a horizontal strip of finite width. For measures in a strip we use a natural generalization of the usual Fourier transform for measures on the line. We consider positive or translation bounded measures μ on a strip whose Fourier transform is a pure point measure μ̂=∑_γ∈Γb_γδ_γ (as usual, δ_γ is the unit mass at the point γ). We prove that the measure ν=∑_γ∈Γ|b_γ|^2δ_γ has the exponential growth. Moreover, if for some η>0 the points of Γ in every interval of length η are linearly independent over integers, then the measure μ̂ also has the exponential growth.
ER -