A free interpolation problem for a subspace of $H^{\mathrm{\infty }}$

Given an inner function $\theta$, the associated star-invariant subspace $K_{\theta }^{\mathrm{\infty }}$ is formed by the functions $f\in H^{\mathrm{\infty }}$ that annihilate (with respect to the usual pairing) the shift-invariant subspace $\theta H^1$ of the Hardy space $H^1$. Assuming that $B$ is an interpolating Blaschke product with zeros $\{a_j\}$, we characterize the traces of functions from $K_B^{\mathrm{\infty }}$ on the sequence $\{a_j\}$. The trace space that arises is, in general, non-ideal (i.e., the sequences $\{w_j\}$ belonging to it admit no nice description in terms of the size of $\mathrm{\mid }{w}_j\mathrm{\mid }$), but we do point out explicit – and sharp – size conditions on $\mathrm{\mid }{w}_j\mathrm{\mid }$ which make it possible to solve the interpolation problem $f(a_j)=w_j$ ($j=1,2,\dots$) with a function $f\in K^\infty_B$.