A model space approach to some classical inequalities for rational functions

2013/10/04 by Anton Baranov, Rachid Zarouf, Baranov, Anton +1 · 1 citation
Mathematics · #FOS: Mathematics #Functional Analysis (math.FA) #Functional Equations Stability Results #Mathematics and Applications

paper · doi:10.48550/arxiv.1310.1182

openalex publication_date 2013/10/04 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28

Abstract

We consider the set R_n of rational functions of degree at most n≥1 with no poles on the unit circle \mathbbT and its subclass R_{n, r} consisting of rational functions without poles in the annulus \ξ: r≤|ξ|≤(1)/(r)\. We discuss an approach based on the model space theory which brings some integral representations for functions in R_n and their derivatives. Using this approach we obtain L^p-analogs of several classical inequalities for rational functions including the inequalities by P. Borwein and T. Erdélyi, the Spijker Lemma and S.M. Nikolskii's inequalities. These inequalities are shown to be asymptotically sharp as n tends to infinity and the poles of the rational functions approach the unit circle \mathbbT.

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