Maximum norm versions of the Szego and Avram-Parter theorems for Toeplitz matrices
Maximum norm versions of the Szego and Avram-Parter theorems for Toeplitz matrices
复制标题
Toeplitz 矩阵的 Szego 和 Avram-Parter 定理的最大范数版本
DOI：
10.1016/j.jat.2015.03.003
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发表时间：
2015
影响因子：
0.9
通讯作者：
Bogoya J
中科院分区：
文献类型：
--
作者：
Bogoya J
Avram-Parter定理描述了大型Toeplitz矩阵奇异值的集体行为。在埃尔米特矩阵的情况下，Avram-Parter定理等价于SzegIgn关于特征值的定理。Avram-Parter定理结合海沟所作的改进，意味着估计符号的奇异值和适当排序的绝对值之间的平均值。本文件有两个目的。在自然假设下，我们首先将已知的均值估计加强到最大范数估计，从而从奇异值的集体结果转向单个奇异值的结果。其次，我们要强调的是，分位数函数的使用大大简化了结果的证明和陈述，并为即将到来的研究提供了一种有前途的语言，为个别奇异值的高阶渐近性。
The collective behavior of the singular values of large Toeplitz matrices is described by the Avram–Parter theorem. In the case of Hermitian matrices, the Avram–Parter theorem is equivalent to Szegő’s theorem on the eigenvalues. The Avram–Parter theorem in conjunction with an improvement made by Trench implies estimates in the mean between the singular values and the appropriately ordered absolute values of the symbol. The purpose of this paper is twofold. Under natural hypotheses, we first strengthen the known estimates in the mean to estimates in the maximum norm, thus turning from collective results on the singular values to results on individual singular values. Secondly, we want to emphasize that the use of the quantile function eases the proofs and statements of results significantly and provides a promising language for forthcoming research into higher order asymptotics for individual singular values.