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arXiv · 2504.03365

Application of methods of quasicrystals theory to entire functions of exponential growth

Sergii Yu. Favorov
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Abstract

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 $\mu$ 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 $\mu$ 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.

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Sergii Yu. Favorov. 2025-04-04. Application of methods of quasicrystals theory to entire functions of exponential growth. https://arxiv.org/abs/2504.03365

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