Application of methods of quasicrystals theory to entire functions of exponential growth
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arXiv
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| Format: | Preprint |
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2025
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| _version_ | 1866912309499133952 |
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| author | Favorov, Sergii Yu. |
| author_facet | Favorov, Sergii Yu. |
| contents | Let $f$ be an entire almost periodic function with zeros in a horizontal strip of finite width; for example, any exponential polynomial with purely imaginary exponents is such a function. Let $μ$ be the measure on the set of zeros of $f$ whose masses coincide with multiplicities of zeros. We define the Fourier transform in the sense of distributions for $μ$ and prove that it is a pure point measure on $\R$ whose complex masses correspond to coefficients of Dirichlet series of the logarithmic derivative of $f$. Bases on this description and Meyer's theorem on quasicrystals, we give a simple necessary and sufficient condition for $f$ to be a finite product of sines. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2504_03365 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | Application of methods of quasicrystals theory to entire functions of exponential growth Favorov, Sergii Yu. Classical Analysis and ODEs Primary42A75 Secondary 42A38, 52C23 Let $f$ be an entire almost periodic function with zeros in a horizontal strip of finite width; for example, any exponential polynomial with purely imaginary exponents is such a function. Let $μ$ be the measure on the set of zeros of $f$ whose masses coincide with multiplicities of zeros. We define the Fourier transform in the sense of distributions for $μ$ and prove that it is a pure point measure on $\R$ whose complex masses correspond to coefficients of Dirichlet series of the logarithmic derivative of $f$. Bases on this description and Meyer's theorem on quasicrystals, we give a simple necessary and sufficient condition for $f$ to be a finite product of sines. |
| title | Application of methods of quasicrystals theory to entire functions of exponential growth |
| topic | Classical Analysis and ODEs Primary42A75 Secondary 42A38, 52C23 |
| url | https://arxiv.org/abs/2504.03365 |