The singular value distribution of the matrix‐sequence {Y_nT_n[f]}_n, with T_n[f] generated by $f\in L^1([-\pi ,\pi ])$, was shown in [J. Pestana and A.J. Wathen, SIAM J Matrix Anal Appl. 2015;36(1):273‐288]. The results on the spectral distribution of {Y_nT_n[f]}_n were obtained independently in [M. Mazza and J. Pestana, BIT, 59(2):463‐482, 2019] and [P. Ferrari, I. Furci, S. Hon, M.A. Mursaleen, and S. Serra‐Capizzano, SIAM J. Matrix Anal. Appl., 40(3):1066‐1086, 2019]. In the latter reference, the authors prove that {Y_nT_n[f]}_n is distributed in the eigenvalue sense as

$${\varphi }_{|f|}(\theta )=\left\{\begin{array}{ll}|f(\theta )|,& \theta \in [0,2\pi ],\\ -|f(-\theta )|,& \theta \in [-2\pi ,0),\end{array}\right.$$

under the assumptions that f belongs to $L^1([-\pi ,\pi ])$ and has real Fourier coefficients. The purpose of this paper is to extend the latter result to matrix‐sequences of the form {h(T_n[f])}_n, where h is an analytic function. In particular, we provide the singular value distribution of the sequence {h(T_n[f])}_n, the eigenvalue distribution of the sequence {Y_nh(T_n[f])}_n, and the conditions on f and h for these distributions to hold. Finally, the implications of our findings are discussed, in terms of preconditioning and of fast solution methods for the related linear systems.

中文翻译：

---

Toeplitz结构函数对称化的大矩阵渐近谱及其在预处理中的应用

矩阵序列{ Y _n T _n [ f ]} _n的奇异值分布，其中T _n [ f ]由$F\in {\mathrm{大号}}^{\mathrm{1个}}（[-\pi ，\pi ]）$，显示在[J. Pestana和AJ Wathen，SIAM J Matrix Anal Appl。2015; 36（1）：273-288]。{ Y _n T _n [ f ]} _n的光谱分布结果是在[M ]中独立获得的。Mazza和J. Pestana，BIT，59（2）：463-482，2019]和[P. Ferrari，I.Furci，S.Hon，MA Mursaleen和S.Serra-Capizzano，SIAM J.Matrix Anal。Appl。，40（3）：1066-1086，2019]。在后一个参考文献中，作者证明{ Y _n T _n [ f ]} _n在特征值意义上分布为

$${\varphi }_{|F|}（\theta ）=\left\{\begin{array}{ll}|F（\theta ）|，& \theta \in [0，2\pi ]，\\ -|F（-\theta ）|，& \theta \in [-2\pi ，0），\end{array}\right.$$

假设f属于${\mathrm{大号}}^{\mathrm{1个}}（[-\pi ，\pi ]）$并具有真实的傅立叶系数 本文的目的是将后者的结果扩展为{ h（T _n [ f ]）} _n形式的矩阵序列，其中h是一个解析函数。特别是，我们提供了序列{ h（T _n [ f ]）} _n的奇异值分布，序列{ Y _n h（T _n [ f ]）} _n的特征值分布以及f和H使这些分布保持不变。最后，从预处理和相关线性系统的快速求解方法的角度讨论了我们的发现的含义。