A class of interpolating Blaschke products and best approximation in \(L^p\) for \(p<1\)
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 \(p_\Theta=1\)?
scientific article; zbMATH DE number 2103484
scientific article