Asymptotics of the L^2 Norm of Derivatives of OPUC

Abstract

We show that for many families of OPUC, one has $||\varphi'_n||_2/n -> 1$, a condition we call normal behavior. We prove that this implies $|\alpha_n| -> 0$ and that it holds if the sequence $\alpha_n$ is in $\ell^1$. We also prove it is true for many sparse sequences. On the other hand, it is often destroyed by the insertion of a mass point.