Orthogonal polynomials on the unit circle: New results

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。

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

We show that the multitude of applications of the Weyl-Titchmarsh m-function leads to a multitude of different functions in the theory of orthogonal polynomials on the unit circle that serve as analogs of the m-function.

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

We prove a general Borg-type result for reflectionless unitary Cantero-Moral-Velazquez (CMV) operators U associated with orthogonal polynomials on the unit circle. The spectrum of U is assumed to be a connected arc on the unit circle. This extends a recent result of Simon in connection with a periodic CMV operator with spectrum the whole unit circle. In the course of deriving the Borg-type result we also use exponential Herglotz representations of Caratheodory functions to prove an infinite sequence of trace formulas connected with the CMV operator U.

Limit-Periodic Verblunsky Coefficients for Orthogonal Polynomials on the Unit Circle

Avila recently introduced a new method for the study of the discrete Schrödinger Operator with limit periodic potential. I adapt this method to the context of orthogonal polynomials in the unit circle with limit periodic Verblunsky Coefficients. Specifically, I represent these Verblunsky Coefficients as a continuous sampling of the orbits of a Cantor group by a minimal translation. I then investigate the measures that arise on the unit circle as I vary the sampling function.

Orthogonal Polynomials on the Unit Circle with quasiperiodic Verblunsky Coefficients have generic purely singular continuous spectrum

As an application of the Gordon lemma for orthogonal polynomials on the unit circle, we prove that for a generic set of quasiperiodic Verblunsky coefficients the corresponding two-sided CMV operator has purely singular continuous spectrum. We use a similar argument to that of the Boshernitzan-Damanik result that establishes the corresponding theorem for the discrete Schrödinger operator.

Weyl-Titchmarsh Theory for CMV Operators Associated with Orthogonal Polynomials on the Unit Circle

We provide a detailed treatment of Weyl-Titchmarsh theory for half-lattice and full-lattice Cantero-Moral-Velazquez (CMV) operators and discuss their systems of orthonormal Laurent polynomials on the unit circle, spectral functions, variants of Weyl--Titchmarsh functions, and Green's functions. In particular, we discuss the corresponding spectral representations of half-lattice and full-lattice CMV operators.

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…

Zeros of non-Baxter paraorthogonal polynomials on the unit circle

We provide leading order asymptotics for the size of the gap in the zeros around 1 of paraothogonal polynomials on the unit circle whose Verblunsky coefficients satisfy a slow decay condition and are inside the interval (-1,0). We also include related results that impose less restrictive conditions on the Verblunsky coefficients.

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…

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…

Rank one perturbations and the zeros of paraorthogonal polynomials on the unit circle

We prove several results about zeros of paraorthogonal polynomials using the theory of rank one perturbations of unitary operators. In particular, we obtain new details on the interlacing of zeros for successive POPUC.

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。

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…

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 …