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 $f$ 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 $\lambda \in [0,1)$.