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Asymptotic sharpness of a Nikolskii type inequality for rational functions in the Wiener algebra

Benjamin Auxemery, Alexander Borichev, Rachid Zarouf

Abstract

We establish the asymptotic sharpness of a Nikolskii type inequality proved by A. Baranov and R. Zarouf for rational functions $f$ in the Wiener algebra of absolutely convergent Fourier series, with at most $n$ poles, all lying outside the dilated disc $\frac{1}λ\mathbb{D}$, where $\mathbb{D}$ denotes the open unit disc and $λ\in[0,1)$ is fixed. More precisely, this inequality tells that the Wiener norm of such functions is bounded by their $H^{2}$-norm -- i.e., their norm in the Hardy space of the disc -- times a factor of order $\sqrt{\frac{n}{1-λ}}$. In this paper, we construct explicit test functions showing that this bound cannot be improved in general: the inequality is asymptotically sharp as $n\to\infty$, up to a universal constant, for every fixed $λ\in[0,1)$.

Asymptotic sharpness of a Nikolskii type inequality for rational functions in the Wiener algebra

Abstract

We establish the asymptotic sharpness of a Nikolskii type inequality proved by A. Baranov and R. Zarouf for rational functions in the Wiener algebra of absolutely convergent Fourier series, with at most poles, all lying outside the dilated disc , where denotes the open unit disc and is fixed. More precisely, this inequality tells that the Wiener norm of such functions is bounded by their -norm -- i.e., their norm in the Hardy space of the disc -- times a factor of order . In this paper, we construct explicit test functions showing that this bound cannot be improved in general: the inequality is asymptotically sharp as , up to a universal constant, for every fixed .
Paper Structure (3 sections, 5 theorems, 93 equations)

This paper contains 3 sections, 5 theorems, 93 equations.

Table of Contents

  1. Introduction
  2. Proof of Theorem \ref{['Nik_type_sharpness']}
  3. Proof of Theorem 3

Key Result

Theorem 1

BZLet $n\ge 1$, $\lambda\in[0,1)$, and let $f\in\mathcal{R}_{n,\,\lambda}$. For some absolute constant $K$ we have

Theorems & Definitions (8)

  • Theorem 1
  • Theorem 2
  • Remark 1
  • Remark 2
  • Theorem 3
  • proof : Proof of Theorem \ref{['Nik_type_sharpness']}
  • Lemma 3.1
  • Theorem 4