Classification Theorems for General Orthogonal Polynomials on the Unit Circle

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{|a_n+j|:0jl}>, n=0,1,2,…,{a_n} 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).     

Multiple Orthogonal Polynomials on the Unit Circle

We introduce multiple orthogonal polynomials on the unit circle. We show how this is related to simultaneous rational approximation to Caratheodory functions (two-point Hermite-Pade approximation near zero and near infinity). We give a Riemann-Hilbert problem for which the solution is in terms of type I and type II multiple orthogonal polynomials on the unit circle, and recurrence relations are obtained from this Riemann-Hilbert problem. Some examples are given to give an idea of the behavior of the zeros of type II multiple orthogonal polynomials.     

Aizenman's Theorem for Orthogonal Polynomials on the Unit Circle

For suitable classes of random Verblunsky coefficients, including independent, identically distributed, rotationally invariant ones, we prove that if $$ \bbE \biggl( \int\f{d\theta}{2\pi} \biggl|\biggl( \f{\calC + e^{i\theta}}{\calC-e^{i\theta}} \biggr)_{k\ell}\biggr|^p \biggr) \leq C_1 e^{-\kappa_1 \abs{k-\ell}} $$ for some $\kappa_1 < 0$ and $p < 1$, then for suitable $C_2$ and $\kappa_2 >0$, $$ \bbE \Bigl( \sup_n \abs{(\calC^n)_{k\ell}}\Bigr) \leq C_2 e^{-\kappa_2 \abs{k-\ell}}. $$ Here $\calC$ is the CMV matrix.     

Asymptotic Behavior of Sobolev-Type Orthogonal Polynomials on the Unit Circle

We study the asymptotic behavior of the sequence of polynomials orthogonal with respect to the discrete Sobolev inner product on the unit circle

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where f(Z)=(f(z₁), …, f^(l₁)(z₁), …, f(z_m), …, f^(l_m)(z_m)), A is a M×M positive definite matrix or a positive semidefinite diagonal block matrix, M=l₁+…+l_m+m, dμ belongs to a certain class of measures, and |z_i|>1, i=1, 2, …, m.     

Mass Points of Measures and Orthogonal Polynomials on the Unit Circle

Orthogonal polynomials on the unit circle are completely determined by their reflection coefficients through the Szeg recurrences. We assume that the reflection coefficients tend to some complex number a with 0<a<1. The orthogonality measure μ then lives essentially on the arc {e^it :αt2π−α} where sin with α(0,π). Under the certain rate of convergence it was proved in (Golinskii et al. (J. Approx. Theory96 (1999), 1–32)) that μ has no mass points inside this arc. We show that this result is sharp in a sense. We also examine the case of the whole unit circle and some examples of singular continuous measures given by their reflection coefficients.     

Discriminants and Functional Equations for Polynomials Orthogonal on the Unit Circle

We derive raising and lowering operators for orthogonal polynomials on the unit circle and find second order differential and q-difference equations for these polynomials. A general functional equation is found which allows one to relate the zeros of the orthogonal polynomials to the stationary values of an explicit quasi-energy and implies recurrences on the orthogonal polynomial coefficients. We also evaluate the discriminants and quantized discriminants of polynomials orthogonal on the unit circle.     

Perturbation of Orthogonal Polynomials on an Arc of the Unit Circle

Orthogonal polynomials on the unit circle are completely determined by their reflection coefficients through the Szeg recurrences. We assume that the reflection coefficients converge to some complex number a with 0 < |a| < 1. The polynomials then live essentially on the are {e^iθ : α ≤ θ ≤ 2 π − α) where cos(α/2) [formula] with α (0, π). We analyze the orthogonal polynomials by comparing them with the orthogonal polynomials with constant reflection coefficients, which were studied earlier by Ya. L. Geronimus and N. I. Akhiezer. In particular, we show that under certain assumptions on the rate of convergence of the reflection coefficients the orthogonality measure will be absolutely continuous on the are. In addition, we also prove the unit circle analogue of M. G. Krein′s characterization of compactly supported nonnegative Borel measures on the real line whose support contains one single limit point in terms of the corresponding system of orthogonal polynomials.     

Ratio Asymptotics for Polynomials Orthogonal on Arcs of the Unit Circle

Ratio asymptotics for orthogonal polynomials on the unit circle is characterized in terms of the existence of lim _n |Φ _n (0)| and {lim _n [ Φ _n+1 (0)/ Φ _n (0)] , where denotes the sequence of reflection coefficients. The limit periodic case, that is, when these limits exist for n = j mod k , j = 1, . . ., k , is also considered. December 27, 1996. Date revised: October 14, 1997. Date accepted: December 22, 1997.     

A Borg-Type Theorem Associated with Orthogonal Polynomials on the Unit Circle

We prove a general Borg-type result for reflectionless unitaryCMV operators U associated with orthogonal polynomials on theunit circle. The spectrum of U is assumed to be a connectedarc on the unit circle. This extends a recent result of Simonin connection with a periodic CMV operator with spectrum thewhole unit circle. In the course of deriving the Borg-type result we also use exponentialHerglotz representations of Caratheodory functions to provean infinite sequence of trace formulas connected with the CMVoperator U.     

A Cubic Decomposition of Sequences of Orthogonal Polynomials on the Unit Circle

In this paper we will discuss the problem of generation of sequences of orthogonal polynomials with respect to measures supported on the unit circle from a given sequence of orthogonal polynomials using a perturbation of a cubic sieved process. The basic tools are the Szeg? forward recurrence relation as well as the fact of the coprimality of orthogonal polynomials on the unit circle and their corresponding reverse polynomials. We also give the connection between the associated orthogonality measures. Finally, some examples of this cubic decomposition are shown.     

Strong Asymptotics for the Continuous Sobolev Orthogonal Polynomials on the Unit Circle

Strong (or Szeg -type) asymptotics for orthogonal polynomials with respect to a Sobolev inner product with general measures (the first measure is arbitrary and the second one is absolutely continuous and satisfying a smoothness condition) is obtained. Examples, illustrating the theorems proved, are presented.     

单位圆周上正交多项式渐近分析的Riemann-Hilbert方法

在单位圆周上考虑带特定权函数的正交多项式,利用Deift P.和Zhou X.所引进的关于振荡型Riemann-Hilbert问题的最速下降法,建立了这类正交多项式在整个复平面上的强渐近公式,发展和改进了一些经典结果.     

单位圆周上正交多项式渐近分析的Riemann-Hilbert方法

在单位圆周上考虑带特定权函数的正交多项式,利用Deift P．和Zhou X．所引进的关于振荡型Riemann-Hilbert问题的最速下降法,建立了这类正交多项式在整个复平面上的强渐近公式,发展和改进了一些经典结果．     

On Uniform Boundedness and Uniform Asymptotics for Orthogonal Polynomials on the Unit Circle

The known conditions due to G. Baxter, Ya. L. Geronimus, and B. L. Golinskii which guarantee the uniform boundedness and/or uniform asymptotic representation for orthonormal polynomials on the unit circle are under consideration. We show that these conditions are in general not necessary. We discuss the relation between the orthonormal polynomials on the unit circle, the best approximations, and absolutely convergent Fourier series.     

Orthogonal Polynomials Satisfying Higher-Order Difference Equations

We introduce a large class of measures with orthogonal polynomials satisfying higher-order difference equations with coefficients independent of the degree of the polynomials. These measures are constructed by multiplying the discrete classical weights of Charlier, Meixner, Krawtchouk, and Hahn by certain variants of the annihilator polynomial of a finite set of numbers.     

Strong Asymptotic Behavior and Weak Convergence of Polynomials Orthogonal on an Arc of the Unit Circle

Let σ be a finite positive Borel measure supported on an arc γ of the unit circle, such that σ′>0 a.e. on γ. We obtain a theorem about the weak convergence of the corresponding sequence of orthonormal polynomials. Moreover, we prove an analogue of the Szeg–Geronimus theorem on strong asymptotics of the orthogonal polynomials on the complement of γ, which completes to its full extent a result of N. I. Akhiezer. The key tool in the proofs is the use of orthogonality with respect to varying measures.     

Cesàro Asymptotics for Orthogonal Polynomials on the Unit Circle and Classes of Measures

The convergence in L²( ) of the even approximants of the Wall continued fractions is extended to the Cesàro–Nevai class CN, which is defined as the class of probability measures σ with lim_n→∞ ∑ⁿ⁻¹_k=0 |a_k|=0, {a_n}_n0 being the Geronimus parameters of σ. We show that CN contains universal measures, that is, probability measures for which the sequence {|_n|² dσ}_n0 is dense in the set of all probability measures equipped with the weak-* topology. We also consider the “opposite” Szeg class which consists of measures with ∑^∞_n=0 (1−|a_n|²)^1/2<∞ and describe it in terms of Hessenberg matrices.     

A Two-Parameter Family of Orthogonal Polynomials with Respect to a Jacobi-Type Weight on the Unit Circle

A two-parameter family of polynomials is introduced by a recursion formula. The polynomials are orthogonal on the unit circle with respect to the weight ω_{α, β}(θ) = |(1 − z)^α(1 + z)^β|², α, β > − , z = e^iθ. Explicit representation, norm estimates, shift identities, and explicit connection to Jacobi polynomials on the real interval [−1, 1] is presented.