Asymptotics of the L2L2 norm of derivatives of OPUC

We show that for many families of OPUC, one has ‖φn′‖2/n→1, a condition we call normal behavior. We prove that this implies |αn|→0|αn|→0 and that it holds if ∑n=0∞|αn|<∞. 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.