Classification Theorems for General Orthogonal Polynomials on the Unit Circle

Abstract

The set P of all probability measures σ on the unit circle T splits into three disjoint subsets depending on properties of the derived set of {|ϕn|2dσ}n⩾0, denoted by Lim(σ). Here {ϕn}n⩾0 are orthogonal polynomials in L2(dσ). The first subset is the set of Rakhmanov measures, i.e., of σ∈P with {m}=Lim(σ), m being the normalized (m(T)=1) Lebesgue measure on T. The second subset Mar(T) consists of Markoff measures, i.e., of σ∈P with m∉Lim(σ), and is in fact the subject of study for the present paper. A measure σ, belongs to Mar(T) iff there are ε>0 and l>0 such that sup{|an+j|:0⩽j⩽l}>ε, n=0,1,2,…,{an} is the Geronimus parameters (=reflectioncoefficients) of σ. We use this equivalence to describe the asymptotic behavior of the zeros of the corresponding orthogonal polynomials (see Theorem G). The third subset consists of σ∈P with {m}⫋Lim(σ). We show that σ is ratio asymptotic iff either σ is a Rakhmanov measure or σ satisfies the López condition (which implies σ∈Mar(T)). Measures σ satisfying Lim(σ)={ν} (i.e., weakly asymptotic measures) are also classified. Either ν is the sum of equal point masses placed at the roots of zn=λ, λ∈T, n=1,2,…, or ν is the equilibrium measure (with respect to the logarithmic kernel) for the inverse image under an m-preserving endomorphism z→zn, n=1,2,…, of a closed arc J (including J=T) with removed open concentric arc J0 (including J0=∅). Next, weakly asymptotic measures are completely described in terms of their Geronimus parameters. Finally, we obtain explicit formulae for the parameters of the equilibrium measures ν and show that these measures satisfy {ν}=Lim(ν).