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<p>In this article, we show that Fourier eigenmeasures supported on spheres with radii given by a locally finite sequence, which we call <i>k</i>-spherical measures, correspond to Fourier series exhibiting a modular-type transformation behaviour with respect to the metaplectic group. A familiar subset of such Fourier series comprises holomorphic modular forms. This allows us to construct <i>k</i>-spherical eigenmeasures and derive Poisson-type summation formulas, thereby recovering formulas of a similar nature established by Cohn–Gonçalves, Lev–Reti, and Meyer, among others. Additionally, we extend our results to higher dimensions, where Hilbert modular forms yield higher-dimensional <i>k</i>-spherical measures.</p>

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Measures, Modular Forms, and Summation Formulas of Poisson Type

  • 期刊论文

  • 出版模式：

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  • 发表日期：

    2025年5月21日

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  • Claudia Alfes,
  • Paul Kiefer,
  • Jan Mazáč

摘要

In this article, we show that Fourier eigenmeasures supported on spheres with radii given by a locally finite sequence, which we call k-spherical measures, correspond to Fourier series exhibiting a modular-type transformation behaviour with respect to the metaplectic group. A familiar subset of such Fourier series comprises holomorphic modular forms. This allows us to construct k-spherical eigenmeasures and derive Poisson-type summation formulas, thereby recovering formulas of a similar nature established by Cohn–Gonçalves, Lev–Reti, and Meyer, among others. Additionally, we extend our results to higher dimensions, where Hilbert modular forms yield higher-dimensional k-spherical measures.