Interpolating sequences for $QA_{B}$

Abstract:

Let $B$ be a closed algebra lying between ${H^\infty }$ and ${L^\infty }$ of the unit circle. We define $QA_B = H^\infty \cap \bar {B}$, the analytic functions in $Q_B = B \cap \bar {B}$. By work of Chang, ${Q_B}$ is characterized by a vanishing mean oscillation condition. We characterize the sequences of points $\left \{{{z_n}} \right \}$ in the open unit disc for which the interpolation problem $f({z_n}) = {\lambda _n}, n = 1, 2,\ldots$, is solvable with $f \in {Q_B}$ for any bounded sequence of numbers $\left \{{{\lambda _n}} \right \}$. Included as a necessary part of our proof is a study of the algebras $Q{A_B}$ and ${Q_B}$.