Banach Spaces of GLT Sequences and Function Spaces

TL;DR

This paper explores the relationship between GLT matrix sequences and function spaces, establishing Banach space and algebra identifications, and applies these results to convergence and preconditioning in PDE discretizations.

Contribution

It extends the theoretical framework connecting GLT sequences with function spaces, providing new Banach space and algebra identifications, and proves convergence equivalences and a Korovkin-type theorem.

Findings

Identified sub-algebras of matrix sequences with sub-algebras of function spaces.

Proved equivalence of eigenvalue/singular value clustering convergence with metric convergence.

Established a Korovkin-type approximation result for GLT sequences.

Abstract

The Generalized Locally Toeplitz (GLT) sequences of matrices have been originated from the study of certain partial differential equations. To be more precise, such matrix sequences arise when we numerically approximate some partial differential equations by discretization. The study of the asymptotic spectral behaviour of GLT sequence is very important in analysing the solution of corresponding partial differential equations. The approximating classes of sequences (a.c.s) and the spectral symbols are important notions in this connection. Recently, G. Barbarino obtained some additional results regarding the theoretical aspects of such notions. He obtained the completeness of the space of matrix sequences with respect to pseudo metric a.c.s. Also, he identified the space of GLT sequences with the space of measurable functions. In this article, we follow the same research line and obtain various results connecting the sub-algebras of matrix sequence spaces and sub-algebras of function spaces. In some cases, these are identifications as Banach spaces and some of them are Banach algebra identifications. In the process, we also prove that the convergence notions in the sense of eigenvalue/singular value clustering are equivalent to the convergence with respect to the metrics introduced here. These convergence notions are related to the study of preconditioners in the case of matrix/operator sequences. Finally, as an application of our main results, we establish a Korovkin-type result in the setting of GLT sequences.