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On the Fourier coefficients of powers of a finite Blaschke product

Alexander Borichev, Karine Fouchet, Rachid Zarouf

Abstract

Given a finite Blaschke product $B$ we prove asymptotically sharp estimates on the $\ell^{\infty}$-norm of the sequence of the Fourier coefficients of $B^{n}$ as $n$ tends to $\infty$. We provide constructive examples which show that our estimates are sharp. As an application we construct a sequence of $n\times n$ invertible matrices $T$ with arbitrary spectrum in the unit disk and such that the quantity $|\det{T}|\cdot\|T^{-1}\|\cdot\|T\|^{1-n}$ grows as a power of $n$. This is motivated by Schäffer's question on norms of inverses.

On the Fourier coefficients of powers of a finite Blaschke product

Abstract

Given a finite Blaschke product we prove asymptotically sharp estimates on the -norm of the sequence of the Fourier coefficients of as tends to . We provide constructive examples which show that our estimates are sharp. As an application we construct a sequence of invertible matrices with arbitrary spectrum in the unit disk and such that the quantity grows as a power of . This is motivated by Schäffer's question on norms of inverses.
Paper Structure (13 sections, 8 theorems, 197 equations)

This paper contains 13 sections, 8 theorems, 197 equations.

Table of Contents

  1. Introduction
  2. Notation and statement of our results
  3. Application to operator theory
  4. Background
  5. Constructive counterexamples to Schäffer's conjecture
  6. Proof of Theorem \ref{['thm:upper_bd']}
  7. Proof of Theorem \ref{['thm:constr_examples']}
  8. The general case $N\ge3$
  9. The case $N=5$
  10. Example 1
  11. Example 2
  12. The case $N=7$
  13. Proof of Theorem \ref{['prop_phi']}

Key Result

Theorem 1

Let $B$ be a finite Blaschke product, and let $\psi_{B}$, $(\xi_\ell)_{\ell=1}^{s}$, $(N_\ell)_{\ell=1}^{s}$ be defined as above. Then we have where $N=\max_{1\le \ell\le s}N_\ell$.

Theorems & Definitions (15)

  • Theorem 1
  • Theorem 2
  • Remark
  • Theorem 3
  • Lemma 4
  • Lemma 5
  • proof : Proof of Theorem \ref{['thm:upper_bd']}.
  • proof : Proof of Theorem \ref{['thm:constr_examples']}, formula \ref{['B']}.
  • Proposition 6
  • proof
  • ...and 5 more