Related papers: The Szego class with a polynomial weight

Let p(t) be a trigonometric polynomial, non-negative on the unit circle. We say that a measure \sigma belongs to a polynomial Szego class, if the logarithm of its density is summable over the circle with the weight p(t). For the associated…

We consider polynomials that are orthogonal over an analytic Jordan curve L with respect to a positive analytic weight, and show that each such polynomial of sufficiently large degree can be expanded in a series of certain integral…

The classical Szego polynomial approximation theorem states that the polynomials are dense in the space $L^2(\rho)$, where $\rho$ is a measure on the unit circle, if and only if the logarithmic integral of the measure $\rho$ diverges. In…

For the polynomials orthogonal on the unit circle with respect to the measure from the Szego class we prove that the polynomial entropy integrals can grow. The estimate obtained is sharp.

In this paper, we extend some classical results of the Szego theory of orthogonal polynomials on the unit circle to the infinite-dimensional case, and we establish the corresponding Szego limit theorem.

Szego's procedure to connect orthogonal polynomials on the unit circle and orthogonal polynomials on [-1,1] is generalized to nonsymmetric measures. It generates the so-called semi-orthogonal functions on the linear space of Laurent…

In this paper we give an asymptotic of the coefficients of the orthogonal polynomials on the unit circle, with respect of a weight of type $\displaystyle{ f : \theta \mapsto \prod_{1\le j \le M} \vert 1 - e^{i(\theta_{j}-\theta)}\vert…

This is an expanded version of the talk given at the conference ``Constructive Functions Tech-04''. We survey some recent results on canonical representation and asymptotic behavior of polynomials orthogonal on the unit circle with respect…

We consider the planar orthogonal polynomial $p_{n}(z)$ with respect to the measure supported on the whole complex plane $${\rm e}^{-N|z|^2} \prod_{j=1}^\nu |z-a_j|^{2c_j}\,{\rm d} A(z)$$ where ${\rm d} A$ is the Lebesgue measure of the…

A classical theorem of Szeg\H{o} states that for any probability measure $\mu=w\frac{\mathrm{d}\theta}{2\pi}+\mu_s$ on the unit circle the polynomials are dense in $L^2(\mathbb{T},\mu)$ if and only if $\log(w)\notin L^1(\mathbb{T})$. A…

Let $E$ be a homogeneous compact set, for instance a Cantor set of positive length. Further let $\sigma$ be a positive measure with $\text{supp}(\sigma)=E$. Under the condition that the absolutely continuous part of $\sigma$ satisfies a…

The main aim of this short paper is to advertize the Koosis theorem in the mathematical community, especially among those who study orthogonal polynomials. We (try to) do this by proving a new theorem about asymptotics of orthogonal…

We prove the following higher-order Szego theorems: if a measure on the unit circle has absolutely continuous part $w(\theta)$ and Verblunsky coefficients $\alpha$ with square-summable variation, then for any positive integer $m$, $\int…

We introduce the Szeg\"o class, Sz(E), for an arbitrary Parreau-Widom set E in R and study the dynamics of its elements under the left shift. When the direct Cauchy theorem holds on C\E, we show that to each J in Sz(E) there is a unique…

We provide a representation in terms of certain canonical functions for a sequence of polynomials orthogonal with respect to a weight that is strictly positive and analytic on the unit circle. These formulas yield a complete asymptotic…

A system of biorthogonal polynomials with respect to a complex valued measure supported on the unit circle is considered and all the terms with bounds are explicitly given for the remainder of an asymptotic formula given by R. Askey for…

Let ${z_n}$ be a sequence in the unit disk ${z\in\mathbb{C}:|z|<1}$. It is known that there exists a unique positive Borel measure in the unit circle ${z\in\mathbb{C}:|z|=1}$ such that the orthogonal polynomials ${\Phi_n}$ satisfy…

Given a measure $\mu$ on the unit sphere $\partial\mathbb{B}^d$ in $\mathbb{C}^d$ with Lebesgue decomposition ${\rm d} \mu = w \, {\rm d} \sigma + {\rm d} \mu_s$, with respect to the rotation-invariant Lebesgue measure $\sigma$ on $\partial…

Let $\mu$ be a measure from Szeg\H{o} class on the unit circle $\mathbb T$ and let $\{f_n\}$ be the family of Schur functions generated by $\mu$. In this paper, we prove a version of the classical Szeg\H{o}'s formula which controls the…

We continue studying polynomials generated by the Szeg\H{o} recursion when a finite number of Verblunsky coefficients lie outside the closed unit disk. We prove some asymptotic results for the corresponding orthogonal polynomials and then…