Rectangular GLT Sequences

TL;DR

This paper extends the theory of generalized locally Toeplitz (GLT) sequences to rectangular matrices, enhancing tools for analyzing spectral distributions in discretized differential problems, with an illustrative application example.

Contribution

It develops a comprehensive theory of rectangular GLT sequences, expanding the existing framework from square matrices to rectangular matrices.

Findings

Enhanced GLT theory for rectangular matrices.

Application example demonstrating potential impact.

Improved spectral analysis tools for discretized differential problems.

Abstract

The theory of generalized locally Toeplitz (GLT) sequences is a powerful apparatus for computing the asymptotic spectral distribution of square matrices $A_n$ arising from the discretization of differential problems. Indeed, as the mesh fineness parameter $n$ increases to $\infty$, the sequence $\{A_n\}_n$ often turns out to be a GLT sequence. In this paper, motivated by recent applications, we further enhance the GLT apparatus by developing a full theory of rectangular GLT sequences as an extension of the theory of classical square GLT sequences. We also detail an example of application as an illustration of the potential impact of the theory presented herein.