Interpolation and duality in spaces of pseudocontinuable functions

TL;DR

This paper characterizes interpolation sequences in spaces of pseudocontinuable functions associated with inner functions, revealing duality relations and non-interpolation phenomena depending on the nature of the inner function.

Contribution

It provides a characterization of interpolation sequences in star-invariant subspaces for various smoothness classes and establishes a new duality relation between specific function spaces.

Findings

Characterization of interpolation sequences in $K^2_B$ for smoothness classes.

Establishment of a non-duality relation between $K^1_ heta$ and $K_{* heta}$.

A non-interpolation result for functions in $K_{*B}$ when $B$ is a Blaschke product.

Abstract

Given an inner function $\theta$ on the unit disk, let $K^p_\theta:=H^p\cap\theta\bar z\bar{H^p}$ be the associated star-invariant subspace of the Hardy space $H^p$. Also, we put $K_{*\theta}:=K^2_\theta\cap{\rm BMO}$. 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 $f\big|_{\mathcal Z}=\mathcal W$ has a solution $f$ in $K^2_B\cap X$. Turning to the case of a general inner function $\theta$, we further establish a non-duality relation between $K^1_\theta$ and $K_{*\theta}$. 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 $K_{*B}$, with $B=B_{\mathcal Z}$ as above.