论文标题
Szegő-Verblunsky类型的定理在多元和几乎周期性的设置中
Theorems of Szegő-Verblunsky type in the multivariate and almost periodic settings
论文作者
论文摘要
经典的szegő-verblunsky定理将概率度量绝对连续部分的对数与圆的概率度量的完全连续部分相关联，以平方列出由该度量确定的正交多发元的复发系数的序列。本文在任意有限维度的圆环上构建正交多项式，以证明在多变量和几乎周期性设置中的Szegő-Verblunsky类型的定理。结果以阻抗形式应用于一维schrödinger方程，以产生有效的新迹线公式，可用于分段恒定阻抗，在这种情况下，经典的痕量公式分解。作为副产品，该分析给出了开放磁盘上有界全态函数的泰勒系数的明确公式。
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 piecewise constant impedance, a case where the classical trace formula breaks down. As a byproduct, the analysis gives an explicit formula for the Taylor coefficients of a bounded holomorphic function on the open disk in terms of its continued fraction expansion.
