Completeness of systems of inner functions

Abstract

For two inner functions $\vartheta ,\ddot{I}†\in H^{\infty }$, we give a simple sufficient condition for the system ${\vartheta }^m,\ddot{I}{†}^n$, $m,n\in \mathbb{Z}$, to be complete in the weak-${}^{\ast }$ topology of $L^{\infty }(\mathbb{T})$. To be precise, we show that this system is complete whenever there is an arc $I$ of the unit circle $\mathbb{T}$ such that $\vartheta$ is univalent on $I$ and $\ddot{I}†$ is univalent on $\mathbb{T}\setminus I$. As an application of this result, we describe a class of analytic curves $\hat{I}“$ such that $(\hat{I}“,\mathcal{X})$ is a Heisenberg uniqueness pair, where $\mathcal{X}$ is the lattice cross $\{(m,n)\in {\mathbb{Z}}^2:mn=0\}$. Our main result extends a theorem of Hedenmalm and Montes-Rodríguez for atomic inner functions with one singularity.