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 } \{ {a_n}\} . The series ∑ ( 1 − | a n | 2 ) / | 1 − ζ ¯ a n | γ \sum {(1 - |{a_n}{|^2})} /|1 - \bar \zeta {a_n}{|^\gamma } , where γ ≧ 1 \gamma \geqq 1 and | ζ | = 1 |\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 {H^2} . Some related conditions and corollaries are also given.