Analogs of the M-Function in the Theory of Orthogonal Polynomials on the Unit Circle

Orthogonal polynomials on the unit circle with Verblunsky coefficients defined by the skew-shift

I give an example of a family of orthogonal polynomials on the unit circle with Verblunsky coefficients given by the skew-shift for which the associated measures are supported on the entire unit circle and almost-every Aleksandrov measure is pure point. Furthermore, I show in the case of the two dimensional skew-shift the zeros of para-orthogonal polynomials obey the same statistics as an appropriate irrational rotation. The proof is based on an analysis of the associated CMV matrices.

A note on the theorems of M. G. Krein and L. A. Sakhnovich on continuous analogs of orthogonal polynomials on the circle

Continuous analogs of orthogonal polynomials on the circle are solutions of a canonical system of differential equations, introduced and studied by M.G.Krein and recently generalized to matrix systems by L.A.Sakhnovich. We prove that the continuous analog of the adjoint polynomials converges in the upper half-plane in the case of L^2 coefficients, but in general the limit can be defined only up to a constant multiple even when the coefficients are in L^p for any p>2, the spectral measure is absolutely continuous and the Szego-Kolmogorov-Krein condition is satisfied. Thus we point out that Krei…

Eigenvalue estimates for non-normal matrices and the zeros of random orthogonal polynomials on the unit circle

We prove that for any $n\times n$ matrix, $A$, and $z$ with $|z|\geq \|A\|$, we have that $\|(z-A)^{-1}\|\leq\cot (\fracπ{4n}) \dist (z, \spec(A))^{-1}$. We apply this result to the study of random orthogonal polynomials on the unit circle.

Limits of Zeros of Orthogonal Polynomials on the Circle

We prove that there is a universal measure on the unit circle such that any probability measure on the unit disk is the limit distribution of some subsequence of the corresponding orthogonal polynomials. This follows from an extension of a result of Alfaro and Vigil (which answered a question of Turán): namely, for $n<N$, one can freely prescribe the $n$-th polynomial and $N-n$ zeros of the $N$-th one. We shall also describe all possible limit sets of zeros within the unit disk.

Schur flows and orthogonal polynomials on the unit circle

The relation between the Toda lattices and similar nonlinear chains and orthogonal polynomials on the real line has been elaborated immensely for the last decades. We examine another system of the differential-difference equations known as the Schur flow within the framework of the theory of orthogonal polynomials on the unit circle. This system can be exhibited in equivalent form as the Lax equation, and the corresponding spectral measure undergoes a simple transformation. The general result is illustrated on the modified Bessel measures on the unit circle and the long time behavior of their …

单位圆上多重正交多项式的Szegő递推

Szegő Recurrence for Multiple Orthogonal Polynomials on the Unit Circle

本文将单位圆上的Szegő递推理论推广到多重正交多项式情形，建立了直接与逆递推、Verblunsky系数类似物及Christoffel–Darboux公式，并与实直线上的最近邻递推形成对应。

Szegő映射与Hermite–Padé多项式用于单位圆上的多重正交性

Szegő Mapping and Hermite--Padé Polynomials for Multiple Orthogonality on the Unit Circle

本文研究了单位圆上满足多重正交条件的多项式，通过Szegő映射和Geronimus关系将其与实线上多重正交多项式联系起来，推导了多项种递推关系和行列式表示。

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 \[ \mathbb{E} \biggl(\int\frac{dθ}{2π} \biggl|\biggl(\frac{\mathcal{C} + e^{iθ}}{\mathcal{C} -e^{iθ}} \biggr)_{k\ell}\biggr|^p \biggr) \leq C_1 e^{-κ_1 |k-\ell|} \] for some $κ_1 >0$ and $p<1$, then for suitable $C_2$ and $κ_2 >0$, \[ \mathbb{E} \bigl(\sup_n |(\mathcal{C}^n)_{k\ell}|\bigr) \leq C_2 e^{-κ_2 |k-\ell|} \] Here $\mathcal{C}$ is the CMV matrix.

Two Variable Orthogonal Polynomials on the Bi-circle and Structured Matrices

We consider bivariate polynomials orthogonal on the bicircle with respect to a positive linear functional. The lexicographical and reverse lexicographical orderings are used to order the monomials. Recurrence formulas are derived between the polynomials of different degrees. These formulas link the orthogonal polynomials constructed using the lexicographical ordering with those constructed using the reverse lexicographical ordering. Relations between the coefficients in the recurrence formulas are derived and used to give necessary and sufficient conditions for the existence of a positive line…

Orthogonal Polynomials with Recursion Coefficients of Generalized Bounded Variation

We consider probability measures on the real line or unit circle with Jacobi or Verblunsky coefficients satisfying an $\ell^p$ condition and a generalized bounded variation condition. This latter condition requires that a sequence can be expressed as a sum of sequences $β^{(l)}$, each of which has rotated bounded variation, i.e., $\sum_{n=0}^\infty | e^{iϕ_l} β_{n+1}^{(l)} - β_n^{(l)} |$ is finite for some $ϕ_l$. This includes discrete Schrödinger operators on a half-line or line with finite linear combinations of Wigner--von Neumann type potentials. For the real line, we prove that in the Leb…

Orthogonal Polynomials on the Unit Circle with Fibonacci Verblunsky Coefficients, II. Applications

We consider CMV matrices with Verblunsky coefficients determined in an appropriate way by the Fibonacci sequence and present two applications of the spectral theory of such matrices to problems in mathematical physics. In our first application we estimate the spreading rates of quantum walks on the line with time-independent coins following the Fibonacci sequence. The estimates we obtain are explicit in terms of the parameters of the system. In our second application, we establish a connection between the classical nearest neighbor Ising model on the one-dimensional lattice in the complex magn…

Perturbations of Orthogonal Polynomials With Periodic Recursion Coefficients

We extend the results of Denisov-Rakhmanov, Szego-Shohat-Nevai, and Killip-Simon from asymptotically constant orthogonal polynomials on the real line (OPRL) and unit circle (OPUC) to asymptotically periodic OPRL and OPUC. The key tool is a characterization of the isospectral torus that is well adapted to the study of perturbations.

Asymptotics of orthogonal polynomials and point perturbation on the unit circle

In the first five sections, we deal with the class of probability measures with asymptotically periodic Verblunsky coefficients of p-type bounded variation. The goal is to investigate the perturbation of the Verblunsky coefficients when we add a pure point to a gap of the essential spectrum. For the asymptotically constant case, we give an asymptotic formula for the orthonormal polynomials in the gap, prove that the perturbation term converges and show the limit explicitly. Furthermore, we prove that the perturbation is of bounded variation. Then we generalize the method to the asymptotically …

Coefficients of Orthogonal Polynomials on the Unit Circle and Higher Order Szego Theorems

Let $μ$ be a non-trivial probability measure on the unit circle $\partial\bbD$, $w$ the density of its absolutely continuous part, $α_n$ its Verblunsky coefficients, and $Φ_n$ its monic orthogonal polynomials. In this paper we compute the coefficients of $Φ_n$ in terms of the $α_n$. If the function $\log w$ is in $L^1(dθ)$, we do the same for its Fourier coefficients. As an application we prove that if $α_n \in \ell^4$ and $Q(z) = \sum_{m=0}^N q_m z^m$ is a polynomial, then with $\bar Q(z) = \sum_{m=0}^N \bar q_m z^m$ and $S$ the left shift operator on sequences we have $|Q(e^{iθ})|^2 \log w(θ…

Measures on the unit circle and unitary truncations of unitary operators

In this paper we obtain new results about the orthogonality measure of orthogonal polynomials on the unit circle, through the study of unitary truncations of the corresponding unitary multiplication operator, and the use of the five-diagonal representation of this operator. Unitary truncations on subspaces with finite co-dimension give information about the derived set of the support of the measure under very general assumptions for the related Schur parameters. Among other results, we obtain a natural generalization of the known result for the Lopez class. On the other hand, unitary truncatio…

单位圆上的Angelesco和AT系统

Angelesco and AT systems on the Unit Circle

本文研究了单位圆上Angelesco和AT系统的Laurent多重正交性，证明了多项式存在性和唯一性，并应用于Hermite-Padé问题的逼近。

Fine structure of the zeros of orthogonal polynomials, III. Periodic recursion coefficients

We discuss asymptotics of the zeros of orthogonal polynomials on the real line and on the unit circle when the recursion coefficients are periodic. The zeros on or near the absolutely continuous spectrum have a clock structure with spacings inverse to the density of zeros. Zeros away from the a.c. spectrum have limit points mod p and only finitely many of them.

OPUC on One Foot

We present an expository introduction to orthogonal polynomials on the unit circle.