单位圆上仿正交多项式的秩一扰动与零点

On the spacing of zeros of paraorthogonal polynomials for singular measures

We prove a lower bound on the spacing of zeros of paraorthogonal polynomials on the unit circle, based on continuity of the underlying measure as measured by Hausdorff dimensions. We complement this with the analog of the result from arXiv:1011.3159 showing that clock spacing holds even for certain singular continuous measures.

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.

Poncelet's Theorem, Paraorthogonal Polynomials and the Numerical Range of Compressed Multiplication Operators

There has been considerable recent literature connecting Poncelet's theorem to ellipses, Blaschke products and numerical ranges, summarized, for example, in the recent book [11]. We show how those results can be understood using ideas from the theory of orthogonal polynomials on the unit circle (OPUC) and, in turn, can provide new insights to the theory of OPUC.

正交多项式与变系数的比值渐进行为及零密度

Ratio asymptotics and zero density for orthogonal polynomials with varying Verblunsky coefficients

本文研究了单位圆上变Verblunsky系数的正交多项式在比值趋于无穷时的渐进行为，推导了比值渐进行为及零分布的结论。

Jordan矩阵的扰动

Perturbations of Jordan Matrices

我们考虑扰动的一个大的约旦矩阵，随机和小的范数或小秩。在这两种情况下，我们表明，大多数的扰动矩阵的特征值是非常接近一个圆的中心在原点。在随机扰动的情况下，我们获得了在某种渐进状态下完全位于圆内的特征值数量的估计。在有限秩扰动的情况下，我们完全确定谱渐近矩阵的大小增加。本文对谱理论的一些标准方法作了初步的说明。