Classification Theorems for General Orthogonal Polynomials on the Unit Circle

Abstract

The set P of all probability measures a on the unit circle T splits into three disjoint subsets depending on properties of the derived set of {\phi(n)\(2) dsigma}(ngreater than or equal to0), denoted by Lim(sigma). Here {phi(n)}(ngreater than or equal to0) are orthogonal polynomials in L-2(dsigma). The first subset is the set of Rakhmanov measures, i.e., of sigma is an element of P with {m} = Lim(sigma), m being the normalized (m(T) = 1) Lebesgue measure on T. The second subset Mar(T) consists of Markoff measures, i.e., of sigma is an element of P with m is not an element of Lim(sigma), and is in fact the subject of study for the present paper. A measure sigma, belongs to Mar(T) iff there are epsilon > 0 and l > 0 such that sup{\a(n+j)\: 0 less than or equal to j less than or equal to l) > epsilon, n = 0, 1, 2,..., {a(n)} is the Geronimus parameters (= reflection coefficients) of sigma. 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 sigma is an element of P with {m} not subset of or equal toLim(sigma). We show that sigma is ratio asymptotic iff either sigma is a Rakhmanov measure or sigma satisfies the Lopez condition (which implies sigma is an element of Mar(T)). Measures sigma satisfying Lim(sigma) = {v} (i.e., weakly asymptotic measures) are also classified. Either v is the sum of equal point masses placed at the roots of z(n) = lambda, lambda is an element of T, n = 1, 2,..., or v is the equilibrium measure (with respect to the logarithmic kernel) for the inverse image under an m-preserving endomorphism z -->z(n), = 1, 2,..., of a closed arc J (including J = T) with removed open concentric are J(0) (including J(0) = empty set). 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 v and show that these measures satisfy {v} = Lim(v). (C) 2002 Elsevier Science (USA).