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<p>We consider a certain uniqueness problem concerning the Fourier coefficients of normalized Cauchy transforms. These problems inherently involve proving a simultaneous approximation phenomenon and establishing the existence of cyclic inner functions in certain sequence spaces. Our results have several applications in different directions. First, we offer a new non-probabilistic proof of a classic theorem by Kahane, Katzenelson and Nestoridis on simultaneous approximation. Secondly, we demonstrate the absence of uniform admissible majorants of Fourier coefficients in de Branges–Rovnyak spaces, and deduce complementary results to the classical Ivashev-Musatov Theorem.</p>

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Fourier coefficients of normalized Cauchy transforms

  • 期刊论文

  • 出版模式：

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  • 发表日期：

    2025年12月3日

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  • Adem Limani

摘要

We consider a certain uniqueness problem concerning the Fourier coefficients of normalized Cauchy transforms. These problems inherently involve proving a simultaneous approximation phenomenon and establishing the existence of cyclic inner functions in certain sequence spaces. Our results have several applications in different directions. First, we offer a new non-probabilistic proof of a classic theorem by Kahane, Katzenelson and Nestoridis on simultaneous approximation. Secondly, we demonstrate the absence of uniform admissible majorants of Fourier coefficients in de Branges–Rovnyak spaces, and deduce complementary results to the classical Ivashev-Musatov Theorem.