Fourier coefficients of normalized Cauchy transforms

TL;DR

This paper investigates the Fourier coefficients of normalized Cauchy transforms, providing new proofs and insights into approximation phenomena and the structure of certain function spaces.

Contribution

It offers a novel non-probabilistic proof of a classical approximation theorem and explores the non-existence of uniform admissible majorants in de Branges-Rovnyak spaces.

Findings

New proof of Kahane and Katzenelson's theorem

Demonstrates absence of uniform admissible majorants in specific spaces

Establishes connections between Fourier coefficients and cyclic inner functions

Abstract

We consider a 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 and Katzenelson on simultaneous approximation. Secondly, we demonstrate the absence of uniform admissible majorants of Fourier coefficients in de Branges-Rovnyak spaces.