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. We define the Fourier transform for measures on a strip that is a natural generalization of the Fourier transform for measures on the line. For positive or translation bounded measures $\mu$ on a strip with the Fourier transform of the form $\hat\mu=\sum\nolimits_{\gamma\in\Gamma}b_\gamma\delta_\gamma$ we prove that the measure $\nu=\sum\nolimits_{\gamma\in\Gamma}|b_\gamma|^2\delta_\gamma$ has the exponential growth. If for some $\eta>0$ the points of $\Gamma$ in every interval of length $\eta$ are linearly independent over integers then the measure $\hat\mu$ also has the exponential growth.
References
L. Alon, M. Kummer, P. Kurasov, C. Vinzant, Higher Dimensional Fourier Quasicrystals from Lee-Yang Varieties, Invent. Math. 239 (2025), 321–376. https://doi.org/10.1007/s00222-024-01307-8
H. Bohr, Almost Periodic Functions, Chelsea Publishing Company, New York, 1951. https://www.amazon.co.uk/Almost-Periodic-Functions-Chelsea-Publishing/dp/082840027X
P. Boyvalenkov, S.Yu. Favorov, Growth of Masses of Crystalline Measures, Studia Math. 290 (2) (2026), 109–119. https://doi.org/10.4064/sm250325-8-1
S.Yu. Favorov, The Crystalline Measure That Is Not a Fourier Quasicrystal, Anal. Math. 50 (2024), 455–462. https://doi.org/10.1007/s10476-024-00031-y
S.Yu. Favorov, Non-negative Crystalline and Poisson Measures in the Euclidean Space, Studia Math. 278 (2024), 81–98. https://doi.org/10.4064/sm240507-2-8
S.Yu. Favorov, Analogues of Fourier Quasicrystals for a Strip, Anal. Math. 51 (2) (2025), 457–475. https://doi.org/10.1007/s10476-025-00089-2
S.Yu. Favorov, Application of Methods of Quasicrystals Theory to Entire Functions of Exponential Growth, Anal. Math. 51 (4) (2025), 1313–1325. https://doi.org/10.1007/s10476-025-00106-4
F. Goncalves, A classification of Fourier summation formulas and crystalline measures, arXiv:2312.11185 (2023). https://arXiv.org/pdf/2312.11185
F. Goncalves, G. Vedana, A complete classification of Fourier summation formulas on the real line, arXiv:2504.02741 (2023). https://arxiv.org/pdf/2504.02741
P. Kurasov, P. Sarnak, Stable Polynomials and Crystalline Measures, J. Math. Phys. 61 (8) (2020), Paper No. 083501. https://doi.org/10.1063/5.0012286
W.M. Lawton, A.K. Tsikh, Fourier Quasicrystals on $mathbb{R}^n$, J. Geom. Anal. (2025), 35–93. https://doi.org/10.1007/s12220-025-01911-x
B.Ya. Levin, Lectures on Entire Functions, Transl. Math. Monogr., vol. 150, Amer. Math. Soc., Providence, RI, 1996. https://bookstore.ams.org/view?ProductCode=MMONO/150
B.M. Levitan, Almost Periodic Functions, Gostehizdat, Moscow, 1953. (in Russian)
Y. Meyer, Global and Local Estimates on Trigonometric Sums, Trans. R. Norw. Soc. Sci. Lett. 2 (2018), 1–25. https://www.dknvs.no/wp-content/uploads/2018/12/Skrifter-2-2018-Meyer-Scr.pdf
Y. Meyer, Multidimensional Crystalline Measures, Trans. R. Norw. Soc. Sci. Lett. 1 (2023), 1–24. https://www.dknvs.no/wp-content/uploads/2023/07/Skrifter-01-2023-Yves-Meyer.pdf
A. Olevskii, A. Ulanovskii, Fourier Quasicrystals with Unit Masses, C. R. Math. Acad. Sci. Paris 358 (2020), 1207–1211. https://doi.org/10.5802/crmath.142
D. Radchenko, M. Viazovska, Fourier Interpolation on the Real Line, Publ. Math. IH´ES 129 (2019), 51–81. https://doi.org/10.1007/s10240-018-0101-z
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