The theory of generalized locally Toeplitz (GLT) sequences is a powerful apparatus for computing the asymptotic spectral distribution of matrices $A_n$ arising from numerical discretizations of differential equations. Indeed, when the mesh fineness parameter $n$ tends to infinity, these matrices $A_n$ give rise to a sequence $\{A_n\}_n$, which often turns out to be a GLT sequence. In this paper, we extend the theory of GLT sequences in several directions: we show that every GLT sequence enjoys a normal form, we identify the spectral symbol of every GLT sequence formed by normal matrices, and we prove that, for every GLT sequence $\{A_n\}_n$ formed by normal matrices and every continuous function $f:\mathbb C\to\mathbb C$, the sequence $\{f(A_n)\}_n$ is again a GLT sequence whose spectral symbol is $f(\kappa)$, where $\kappa$ is the spectral symbol of $\{A_n\}_n$. In addition, using the theory of GLT sequences, we prove a spectral distribution result for perturbed normal matrices.

翻译：广义局部Toeplitz（GLT）序列理论是计算微分方程数值离散化产生的矩阵$A_n$渐近谱分布的有力工具。事实上，当网格细度参数$n$趋于无穷时，这些矩阵$A_n$生成序列$\{A_n\}_n$，而该序列通常为GLT序列。本文从多个方向扩展了GLT序列理论：证明每个GLT序列都具有正规形式，识别由正规矩阵构成的每个GLT序列的谱符号，并证明对于由正规矩阵构成的任意GLT序列$\{A_n\}_n$及任意连续函数$f:\mathbb C\to\mathbb C$，序列$\{f(A_n)\}_n$仍是GLT序列，且其谱符号为$f(\kappa)$，其中$\kappa$为$\{A_n\}_n$的谱符号。此外，利用GLT序列理论，我们证明了扰动正规矩阵的一个谱分布结果。