@TechReport{ it:2015-023,
author = {Carlo Garoni and Stefano Serra-Capizzano},
title = {The theory of {G}eneralized {L}ocally {T}oeplitz
sequences: a review, an extension, and a few representative
applications},
institution = {Department of Information Technology, Uppsala University},
department = {Division of Scientific Computing},
year = {2015},
number = {2015-023},
month = aug,
note = {Revised, corrected, updated and extended version of TR
2015-016.},
abstract = {We review and extend the theory of Generalized Locally
Toeplitz (GLT) sequences, which goes back to the pioneering
work by Tilli on Locally Toeplitz (LT) sequences and was
developed by the second author during the last decade.
Informally speaking, a GLT sequence $\{A_n\}_n$ is a
sequence of matrices with increasing size, equipped with a
Lebesgue-measurable function $\kappa$ (the so-called
symbol). This function characterizes the asymptotic
singular value distribution of $\{A_n\}_n$; in the case
where the matrices $A_n$ are Hermitian, it also
characterizes the asymptotic eigenvalue distribution of
$\{A_n\}_n$. Three fundamental examples of GLT sequences
are: (i) the sequence of Toeplitz matrices generated by a
function $f$ in $L^1$; (ii) the sequence of diagonal
sampling matrices containing the evaluations of a
Riemann-integrable function $a$ over a uniform grid; (iii)
any zero-distributed sequence, i.e., any sequence of
matrices possessing an asymptotic singular value
distribution characterized by the identically zero
function. The symbol of the GLT sequence (i) is $f$, the
symbol of the GLT sequence (ii) is $a$, and the symbol of
any GLT sequence of the form (iii) is 0. The set of GLT
sequences is a *-algebra. More precisely, suppose that
$\{A_n^{(1)}\}_n,\ldots,\{A_n^{(r)}\}_n$ are GLT sequences
with symbols $\kappa_1,\ldots,\kappa_r$, and let
$A_n=\textup{ops}(A_n^{(1)},\ldots,A_n^{(r)})$ be a matrix
obtained from $A_n^{(1)},\ldots,A_n^{(r)}$ by means of
certain algebraic operations `ops', such as linear
combinations, products, inversions and Hermitian
transpositions; then, $\{A_n\}_n$ is a GLT sequence with
symbol $\kappa=\textup{ops}(\kappa_1,\ldots,\kappa_r)$.
As already proved in several contexts, the theory of GLT
sequences is a powerful apparatus for computing the
asymptotic singular value and eigenvalue distribution of
the discretization matrices $A_n$ arising from the
numerical approximation of continuous problems, such as
integral equations and, especially, partial differential
equations. Indeed, when the discretization parameter $n$
tends to infinity, the discretization matrices $A_n$ give
rise to a sequence $\{A_n\}_n$, which often turns out to be
a GLT sequence.
However, in this work we are not concerned with the
applicative interest of the theory of GLT sequences, for
which we limit to outline some of the numerous applications
and to refer the reader to the available literature. On the
contrary, we focus on the mathematical foundations. We
propose slight (but relevant) modifications of the original
definitions, and we introduce for the first time the
concept of LT sequences in the multivariate/multilevel
setting. With the new definitions, based on the notion of
approximating class of sequences, we are able to enlarge
the applicability of the theory, by generalizing and/or
simplifying a lot of key results. In particular, we remove
a technical hypothesis concerning the Riemann-integrability
of the so-called `weight functions', which appeared in the
statement of many spectral distribution and algebraic
results for GLT sequences. Moreover, we provide a formal
and detailed proof of the fact that the sequences of
matrices mentioned in items (i)--(iii) fall in the class of
LT sequences. Several versions of this result were already
present in previous papers, but only partial proofs were
given.
As a final step, we extend the theory of GLT sequences. We
first prove an approximation result, which is particularly
useful to show that a given sequence of matrices is a GLT
sequence. By using this result, we provide a new and easier
proof of the fact that $\{A_n^{-1}\}_n$ is a GLT sequence
with symbol $\kappa^{-1}$ whenever $\{A_n\}_n$ is a GLT
sequence of invertible matrices with symbol $\kappa$ and
$\kappa\ne 0$ almost everywhere. Finally, using again the
approximation result, we prove that $\{f(A_n)\}_n$ is a GLT
sequence with symbol $f(\kappa)$, as long as
$f:\mathbb{R}\to\mathbb{R}$ is continuous and $\{A_n\}_n$
is a GLT sequence of Hermitian matrices with symbol
$\kappa$. This has important implications, e.g., in proving
that the geometric mean of two GLT sequences is still a GLT
sequence, with symbol given by the the geometric mean of the symbols.}
}