Logo
ScienceStack
Table of Contents
Fetching ...
Sign In

Fourier coefficients of normalized Cauchy transforms

Adem Limani

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.

Fourier coefficients of normalized Cauchy transforms

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.

Paper Structure

This paper contains 19 sections, 19 theorems, 68 equations.

Table of Contents

  1. Introduction
  2. Normalized Cauchy transforms
  3. Main results
  4. Fourier coefficients of normalized Cauchy transforms
  5. Cyclic inner functions
  6. Simultaneous approximation and removable sets
  7. Uniformly admissible majorants
  8. Organization and notation
  9. Cauchy dual reformulations
  10. Orlicz spaces and duality
  11. The role of de Branges-Rovnyak spaces
  12. Cyclic inner functions in $\mathcal{B}_0(w)$
  13. Sharp Control
  14. A criterion for cyclicity
  15. Orlicz embeddings of Bloch spaces
  16. ...and 4 more sections

Key Result

Theorem 1.1

For any positive finite Borel measure $\mu$ on $\mathbb{T}$, the image $\mathcal{K}_{\mu}L^1(\mu)$ contains functions $f$ with uniformly convergent Fourier series on $\mathbb{T}$. In fact, the set of such functions form a dense subspace in the corresponding de Branges-Rovnyak space.

Theorems & Definitions (31)

  • Theorem 1.1: See limani2022abstract
  • Theorem 2.1
  • Corollary 2.2
  • Theorem 2.3
  • Theorem 2.4
  • Theorem 2.5
  • Corollary 2.6: Kahane, Nestoridis, kahane2000series
  • Theorem 2.7
  • Theorem 2.8
  • Lemma 3.1
  • ...and 21 more