A class of interpolating Blaschke products and best approximation in \(L^p\) for \(p<1\)
type: Item
instance of: scholarly article
title: A class of interpolating Blaschke products and best approximation in \(L^p\) for \(p<1\)
zbMATH Open document ID: 1055.30030
DOI: 10.1007/BF03321865
published in: Computational Methods and Function Theory
publication date: 2004-09-27T00:00:00Z
review text: Let \(\Theta\) be an inner function and denote by \(H^p\), \(0<p<\infty\), the usual Hardy space on the unit disk. The author considers the problem of describing the distance of the constant function 1 to the space \(\Theta H^p\). So let \(\varepsilon_p(\Theta)=\inf \{\| f\| ^p_p: f\in 1+\Theta H^p\}\). Since \((1-\overline{\Theta(0)}\Theta)^{2/p}\) belongs to \(1+\Theta H^\infty\), one gets that \(\varepsilon_p(\Theta)\leq 1-| \Theta(0)| ^2\) for all \(p>0\). Moreover, it is known that \(\varepsilon_p(\Theta)=1-| \Theta(0)| ^2\) for \(p\geq 1\) and \(\varepsilon_p(\Theta)=1\) for all \(p>0\) if \(\Theta(0)=0\). On the other hand, let \(p_\Theta=\inf\{p>0: \varepsilon_p(\Theta)=1-| \Theta(0)| ^2\}\). Then \(p_\Theta\) may be strictly positive: indeed, any inner function \(\Theta\) with \(\Theta(0)\not=0\) having the property that there exists an other inner function \(I\) that is a proper divisor of \(\Theta\) but for which \(\Theta\) divides \(I^2\), is such an example. The author now studies inner functions \(\Theta\), \(\Theta(0)\not=0\), for which \(p_\Theta=0\). One of his main results is that any such function necessarily is an interpolating Blaschke product. The author also investigates the functional \(\varepsilon_p^0(\Theta):=\{\| f\| ^p_p: f\in 1+\Theta H^p\), \(f\not=0\) in \({\mathbb D}\}\) and shows that the set of inner functions \(\Theta\) for which \(\varepsilon_p^0(\Theta)=1-| \Theta(0)| ^2\) for all \(p>0\) contains a singular inner function. The paper concludes with many interesting open questions: e.g., does there exist an inner function \(\Theta\) for which \(p_\Theta=1\)?
reviewed by: Raymond Mortini
Mathematics Subject Classification ID: 30D50
Mathematics Subject Classification ID: 30D55
zbMATH DE Number: 2103484
zbMATH Keywords: inner functions
zbMATH Keywords: approximation
zbMATH Keywords: Hardy spaces
author: A. B. Aleksandrov
MaRDI profile type: Publication
cites work: Q3924466
cites work: Q3922907
cites work: Q5510566
full work available at URL: https://doi.org/10.1007/bf03321865
OpenAlex ID: W1972616809
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description: scientific article; zbMATH DE number 2103484