Classification Theorems for General Orthogonal Polynomials on the Unit Circle |
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Authors: | S. V. Khrushchev |
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Affiliation: | Department of Mathematics, Atilim University, Incek, 06836, Ankara, Turkeyf1 |
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Abstract: | The set of all probability measures σ on the unit circle splits into three disjoint subsets depending on properties of the derived set of {|n|2dσ}n0, denoted by Lim(σ). Here {n}n0 are orthogonal polynomials in L2(dσ). The first subset is the set of Rakhmanov measures, i.e., of σ with {m}=Lim(σ), m being the normalized (m()=1) Lebesgue measure on . The second subset Mar() consists of Markoff measures, i.e., of σ with mLim(σ), and is in fact the subject of study for the present paper. A measure σ, belongs to Mar() iff there are >0 and l>0 such that sup{|an+j|:0jl}>, 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 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 σMar()). 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, 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=) 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). |
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