Asymptotics of the image norm of derivatives of OPUC Original Research Article

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