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<p>Let <i>f</i> 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 <InlineEquation ID="IEq1"> <EquationSource Format="TEX">\(\mu\)</EquationSource> <EquationSource Format="MATHML"><math> <mi>μ</mi> </math></EquationSource> </InlineEquation> be the measure on the set of zeros of <i>f </i> whose masses coincide with multiplicities of zeros. We define the Fourier transform in the sense of distributions for <InlineEquation ID="IEq10"> <EquationSource Format="TEX">\(\mu\)</EquationSource> <EquationSource Format="MATHML"><math> <mi>μ</mi> </math></EquationSource> </InlineEquation> and prove that it is a pure point measure on <InlineEquation ID="IEq100"> <EquationSource Format="TEX">\(\mathbb{R}\)</EquationSource> <EquationSource Format="MATHML"><math> <mi mathvariant="double-struck">R</mi> </math></EquationSource> </InlineEquation> whose complex masses correspond to coefficients of Dirichlet series of the logarithmic derivative of <i>f .</i> Bases on this description and Meyer’s theorem on quasicrystals, we give a simple necessary and sufficient condition for <i>f</i> to be a finite product of sines.</p>

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

  • 期刊论文

  • 发表日期：

    2025年9月3日

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  • S. Yu. Favorov

摘要

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 \(\mu\) 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 \(\mu\) μ and prove that it is a pure point measure on \(\mathbb{R}\) 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.