Paper
Asymptotic sharpness of a Nikolskii type inequality for rational functions in the Wiener algebra
Authors
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 $λ\in[0,1)$.
Metadata
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Raw Data (Debug)
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"raw_xml": "<entry>\n <id>http://arxiv.org/abs/2603.03908v1</id>\n <title>Asymptotic sharpness of a Nikolskii type inequality for rational functions in the Wiener algebra</title>\n <updated>2026-03-04T10:14:03Z</updated>\n <link href='https://arxiv.org/abs/2603.03908v1' rel='alternate' type='text/html'/>\n <link href='https://arxiv.org/pdf/2603.03908v1' rel='related' title='pdf' type='application/pdf'/>\n <summary>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 $λ\\in[0,1)$.</summary>\n <category scheme='http://arxiv.org/schemas/atom' term='math.CA'/>\n <published>2026-03-04T10:14:03Z</published>\n <arxiv:primary_category term='math.CA'/>\n <author>\n <name>Benjamin Auxemery</name>\n <arxiv:affiliation>I2M</arxiv:affiliation>\n </author>\n <author>\n <name>Alexander Borichev</name>\n <arxiv:affiliation>I2M</arxiv:affiliation>\n </author>\n <author>\n <name>Rachid Zarouf</name>\n <arxiv:affiliation>ADEF, CPT</arxiv:affiliation>\n </author>\n </entry>"
}