On the Accumulation of the Zeros of a Blaschke Product at a Boundary Point

Abstract

Let B be a Blaschke product with zeros $\{ {a_n}\}$. The series $\sum {(1 - |{a_n}{|^2})} /|1 - \bar \zeta {a_n}{|^\gamma }$, where $\gamma \geqq 1$ and $|\zeta | = 1$, has been used by P. R. Ahern, D. N. Clark, G. T. Cargo, and others in the study of the boundary behavior of B and various associated functions. In this paper the convergence of this series is shown to be equivalent to a condition on a reproducing kernel for a subspace of the Hardy space ${H^2}$. Some related conditions and corollaries are also given.