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

TL;DR

This paper proves that a specific inequality relating Wiener algebra norms and Hardy space norms for rational functions with poles outside a dilated disk is asymptotically optimal, using explicit test functions.

Contribution

It establishes the asymptotic sharpness of a Nikolskii type inequality for rational functions in the Wiener algebra with poles outside a fixed dilated disk.

Findings

The inequality's bound cannot be improved asymptotically.

Explicit test functions demonstrate the sharpness of the inequality.

The bound involves the square root of the ratio of the degree to (1 - lambda).

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 $\frac{1}{\lambda}\mathbb{D}$, where $\mathbb{D}$ denotes the open unit disc and $\lambda\in[0,1)$ 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 $\sqrt{\frac{n}{1-\lambda}}$. 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 $\lambda\in[0,1)$.