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<p>We consider the Poincaré theta series operator associated with a discrete subgroup in the complex plane. We show that this operator is onto for certain appropriate spaces where it is well-defined. A concrete description of its kernel is provided, expressed in terms of special functions and dependent on the rank of the discrete subgroup. Additionally, we present explicit conditions on the Fourier coefficients necessary for a function to belong to this kernel.</p>

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On the planar Poincaré theta series

  • 期刊论文

  • 发表日期：

    2025年9月22日

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  • Aymane El Fardi,
  • Allal Ghanmi,
  • Lahcen Imlal,
  • Mohammed Ziyat

摘要

We consider the Poincaré theta series operator associated with a discrete subgroup in the complex plane. We show that this operator is onto for certain appropriate spaces where it is well-defined. A concrete description of its kernel is provided, expressed in terms of special functions and dependent on the rank of the discrete subgroup. Additionally, we present explicit conditions on the Fourier coefficients necessary for a function to belong to this kernel.