论文标题
特殊正交多项式的特殊零的收敛速率
Convergence Rates of Exceptional Zeros of Exceptional Orthogonal Polynomials
论文作者
论文摘要
我们考虑了异常正交多项式（XOP）的零。特殊的正交多项式最初被发现为经典Bochner-Brenke分类以外的二阶差分算子的特征函数，因为XOP序列忽略了某些程度的多项式。这种遗漏导致经典正交多项式序列的几种特性不扩展到XOP序列。这样的属性之一就是将零对正交度度量的支持的凸面进行限制。在XOP情况下，在经典间隔之外存在的零被称为异常零，并且随着度变大，它们通常会收敛到易于识别的极限点。我们推断出融合的确切速率，并验证先前出现在文献中的某些估计值是否敏锐。
We consider the zeros of exceptional orthogonal polynomials (XOP). Exceptional orthogonal polynomials were originally discovered as eigenfunctions of second order differential operators that exist outside the classical Bochner-Brenke classification due to the fact that XOP sequences omit polynomials of certain degrees. This omission causes several properties of the classical orthogonal polynomial sequences to not extend to the XOP sequences. One such property is the restriction of the zeros to the convex hull of the support of the measure of orthogonality. In the XOP case, the zeros that exist outside the classical intervals are called exceptional zeros and they often converge to easily identifiable limit points as the degree becomes large. We deduce the exact rate of convergence and verify that certain estimates that previously appeared in the literature are sharp.
