Generalized Locally Toeplitz (GLT) matrix sequences arise from large linear systems that approximate Partial Differential Equations (PDEs), Fractional Differential Equations (FDEs), and Integro-Differential Equations (IDEs). GLT sequences of matrices have been developed to study the spectral/singular value behaviour of the numerical approximations to various PDEs, Fades and IDEs. These approximations can be achieved using any discretization method on appropriate grids through local techniques such as Finite Differences, Finite Elements, Finite Volumes, Isogeometric Analysis, and Discontinuous Galerkin methods. Spectral and singular value symbols are essential for analyzing the eigenvalue and singular value distributions of matrix sequences in the Weyl sense. In this article, we provide a comprehensive overview of the operator-theoretic aspect of GLT sequences. The theory of GLT sequences, along with findings on the asymptotic spectral distribution of perturbed matrix sequences, is a highly effective and successful method for calculating the spectral symbol f. Therefore, developing an automatic procedure to compute the spectral symbols of these matrix sequences would be advantageous, a task that Ahmed Ratnani, N S Sarathkumar, S. Serra-Capizzano have partially undertaken. As an application of the theory developed here, we propose an automatic procedure for computing the symbol of the underlying sequences of matrices, assuming they form a GLT sequence that meets mild conditions.

翻译：广义局部Toeplitz（GLT）矩阵序列源自逼近偏微分方程（PDE）、分数阶微分方程（FDE）及积分微分方程（IDE）的大型线性系统。GLT矩阵序列的发展旨在研究各类PDE、FDE和IDE数值逼近的谱/奇异值特性。这些逼近可通过有限差分法、有限元法、有限体积法、等几何分析及间断伽辽金法等局部化技术，在适当网格上采用任意离散化方法实现。谱符号与奇异值符号对于分析矩阵序列在Weyl意义下的特征值与奇异值分布至关重要。本文系统阐述了GLT序列的算子理论层面。GLT序列理论结合扰动矩阵序列渐近谱分布的研究成果，已成为计算谱符号f的高效且成功的方法。因此，建立计算此类矩阵序列谱符号的自动化流程具有重要价值——Ahmed Ratnani、N S Sarathkumar与S. Serra-Capizzano已在此方向取得部分进展。基于本文发展的理论，我们提出一种自动化计算矩阵序列符号的流程，其前提是该序列构成满足温和条件的GLT序列。