Fourier coefficients of normalized Cauchy transforms

Abstract 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.