Growth of masses of crystalline measures
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arXiv
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| Format: | Preprint |
| Published: |
2025
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| _version_ | 1866912293191680000 |
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| author | Boyvalenkov, Peter Favorov, Sergii Yu. |
| author_facet | Boyvalenkov, Peter Favorov, Sergii Yu. |
| contents | Let $μ$ be a measure on the Euclidean space $\R^d$ of unbounded total variation that is positive or translation bounded and has a pure point Fourier transform in the sense of distributions $\hatμ$. We prove that the measure $ν$ with the same support as $\hatμ$ and masses equal to the squares of the masses of $\hatμ$ is translation bounded. We also prove that if $μ$ is as above and the restriction of its spectrum, i.e., of the support of $\hatμ$, to each ball of fixed radius is a linearly independent set over $\Z$, then the measure $\hatμ$ is also translation bounded. These results imply certain conditions for a crystalline measure to be a Fourier quasicrystal. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2503_19567 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | Growth of masses of crystalline measures Boyvalenkov, Peter Favorov, Sergii Yu. Functional Analysis 42A38, 42A75, 52C23 Let $μ$ be a measure on the Euclidean space $\R^d$ of unbounded total variation that is positive or translation bounded and has a pure point Fourier transform in the sense of distributions $\hatμ$. We prove that the measure $ν$ with the same support as $\hatμ$ and masses equal to the squares of the masses of $\hatμ$ is translation bounded. We also prove that if $μ$ is as above and the restriction of its spectrum, i.e., of the support of $\hatμ$, to each ball of fixed radius is a linearly independent set over $\Z$, then the measure $\hatμ$ is also translation bounded. These results imply certain conditions for a crystalline measure to be a Fourier quasicrystal. |
| title | Growth of masses of crystalline measures |
| topic | Functional Analysis 42A38, 42A75, 52C23 |
| url | https://arxiv.org/abs/2503.19567 |