Computational Methods and Function Theory 4 (2004), No. 2, 315--326
Copyright Heldermann Verlag 2004
Alexei B. Aleksandrov
alex@pdmi.ras.ru , St. Petersburg Branch, Steklov Mathematical Institute, Russian Academy of Sciences, Fontanka, 27, 191023, St. Petersburg, Russia.
A function $u$ on the unit circle $\mathbb{T}$ is said to be badly approximable in the weighted space $L^p(\mathbb{T},w)$ if $\|u+f\|_{L^p(\mathbb{T},w)}\ge\|u\|_{L^p(\mathbb{T},w)}$ for all $f\in H^\infty$. We prove that if an unimodular function $u$ is badly approximable in $L^p(\mathbb{T},w)$ for all $p\in(0,+\infty)$ and some non-zero weight $w$, then $\overline{u}$ is an inner function. We describe the inner functions $\Theta$ and the weights $w$ on the unit circle $\mathbb{T}$ such that $\overline{\Theta}$ is badly approximable in $L^p(\mathbb{T},w)$ for all $p>0$. It turns out that, for given inner functions $\Theta$, the class of all weights satisfying the above-mentioned condition depends only on the zero set of $\Theta$. In other words, $\overline{\Theta}$ is badly approximable in $L^p(\mathbb{T},w)$ for all $p\in(0,+\infty)$ if and only if $\overline{B}$ is badly approximable in $L^p(\mathbb{T},w)$ for all $p\in(0,+\infty)$, where $B$ is a Blaschke product with simple zeros and such that $\Theta^{-1}(0)=B^{-1}(0)$.
Keywords: Hardy spaces, inner functions, best approximation.
MSC 2000: 30D50, 30D55.
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