Two Counterexamples Involving Inner Functions
Kenneth Stephenson · Proceedings of the American Mathematical Society · 1979
Two questions about the existence of bounded analytic functions with prescribed behavior are answered using a construction technique of McLaughlin and Piranian. If A is a relatively closed, countable subset of the unit disc, the construction gives (I) Function $f \in {H^\infty }$ so that the inner factor of $f - \alpha$ is a finite Blaschke product if and only if $\alpha \in A$. (II) Inner function $\phi$ so that $\phi - \alpha$ has a discrete singular inner factor if and only if $\alpha \in A$.