Asymptotics of the L^2 Norm of Derivatives of OPUC

TL;DR

This paper investigates the asymptotic behavior of the L^2 norms of derivatives of orthogonal polynomials on the unit circle (OPUC), establishing conditions under which they exhibit normal behavior and analyzing factors that influence this property.

Contribution

It introduces the concept of normal behavior for OPUC derivatives, proves its relation to the decay of Verblunsky coefficients, and explores how sparse sequences and mass points affect this behavior.

Findings

||φ'_n||_2/n approaches 1 for many OPUC families

Normal behavior implies α_n tends to zero

Sparse sequences often exhibit normal behavior

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.