Interpolation and duality in spaces of pseudocontinuable functions

Abstract

Given an inner function $\theta$ on the unit disk, let $K_{\theta }^p:=H^p\cap \theta \bar{z}\overline{H^p}$ be the associated star-invariant subspace of the Hardy space $H^p$. Also, we put $K_{\ast \theta }:=K_{\theta }^2\cap \mathrm{BMO}$. Assuming that $B=B_{\mathcal{Z}}$ is an interpolating Blaschke product with zeros $\mathcal{Z}=\{z_j\}$, we characterize, for a number of smoothness classes $X$, the sequences of values $\mathcal{W}=\{w_j\}$ such that the interpolation problem $f{|}_{\mathcal{Z}}=\mathcal{W}$ has a solution $f$ in $K_B^2\cap X$. Turning to the case of a general inner function $\theta$, we further establish a non-duality relation between $K_{\theta }^1$ and $K_{\ast \theta }$. Namely, we prove that the latter space is properly contained in the dual of the former, unless $\theta$ is a finite Blaschke product. From this we derive an amusing non-interpolation result for functions in $K_{\ast B}$, with $B=B_{\mathcal{Z}}$ as above.

12 pages

References in corpus (3)

• Duality theorems for coinvariant subspaces of $H^1$
• A free interpolation problem for a subspace of $H^{\infty }$
• Interpolating by functions from model subspaces in $H^1$