Bernstein-Szegő measures on the two dimensional torus

Orthogonality relations for bivariate Bernstein-Szegő measures

The orthogonality properties of certain subspaces associated with bivariate Bernstein-Szegő measures are considered. It is shown that these spaces satisfy more orthogonality relations than expected from the relations that define them. The results are used to prove a Christoffel-Darboux like formula for these measures.

平面上的Bernstein-Szegő测度

Bernstein-Szegő measures in the plane

本文定义了一类平面上的Bernstein-Szegő测度，利用Fejér-Riesz分解和Szegő映射的二维推广，建立了其谱性质并给出了有限矩刻画。

Analytic scattering theory for Jacobi operators and Bernstein-Szegö asymptotics of orthogonal polynomials

We study semi-infinite Jacobi matrices $H=H_{0}+V$ corresponding to trace class perturbations $V$ of the "free" discrete Schrödinger operator $H_{0}$. Our goal is to construct various spectral quantities of the operator $H$, such as the weight function, eigenfunctions of its continuous spectrum, the wave operators for the pair $H_{0}$, $H$, the scattering matrix, the spectral shift function, etc. This allows us to find the asymptotic behavior of the orthonormal polynomials $P_{n}(z)$ associated to the Jacobi matrix $H $ as $n\to\infty$. In particular, we consider the case of $z$ inside the spe…

Bernstein-Markov测度与Toeplitz理论

Bernstein-Markov measures and Toeplitz theory

本文证明在赋予大线丛的紧复流形上，与Bernstein-Markov测度关联的Toeplitz算子在复合下构成代数，并基于Bergman核非对角渐近分析推导出Szegő型谱等分布结果。

Theorems of Szegő-Verblunsky type in the multivariate and almost periodic settings

The classical Szegő-Verblunsky theorem relates integrability of the logarithm of the absolutely continuous part of a probability measure on the circle to square summability of the sequence of recurrence coefficients for the orthogonal polynomials determined by the measure. The present paper constructs orthogonal polynomials on the torus of arbitrary finite dimension in order to prove theorems of Szegő-Verblunsky type in the multivariate and almost periodic settings. The results are applied to the one-dimensional Schrödinger equation in impedance form to yield a new trace formula valid for piec…

多维环面上的第一 Szegő 极限定理

The first Szegő limit theorem on multi-dimensional torus

本文证明了多维环面上第一 Szegő 极限定理，包括有限维和无限维情形，并解决了 Nikolski 和 Pushnitski 提出的两个问题。