@TechReport{ it:2015-016,
author = {Carlo Garoni and Stefano Serra-Capizzano},
title = {{G}eneralized {L}ocally {T}oeplitz sequences: a review and
an extension},
institution = {Department of Information Technology, Uppsala University},
department = {Division of Scientific Computing},
year = {2015},
number = {2015-016},
month = may,
note = {Revised, corrected, updated and extended by TR 2015-023
(\url{http://www.it.uu.se/research/publications/reports/2015-023}).}
,
abstract = {We review the theory of Generalized Locally Toeplitz (GLT)
sequences, hereinafter called `the GLT theory', 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: every GLT sequence has a measurable
symbol; the singular value distrbution of any GLT sequence
is identified by the symbol (also the eigenvalue
distribution if the sequence is made by Hermitian
matrices); the GLT sequences form an algebra, closed under
linear combinations, (pseudo)-inverse if the symbol
vanishes in a set of zero measure, product and the symbol
obeys to the same algebraic manipulations.
As already proved in several contexts, this theory is a
powerful tool for computing/analyzing the asymptotic
spectral distribution of the discretization matrices
arising from the numerical approximation of continuous
problems, such as Integral Equations and, especially,
Partial Differential Equations, including variable
coefficients, irregular domains, different approximation
schemes such as Finite Differences, Finite Elements,
Collocation/Galerkin Isogeometric Analysis etc.
However, in this review we are not concerned with the
applicative interest of the GLT theory, for which we limit
to refer the reader to the numerous applications available
in the literature.
On the contrary, we focus on the theoretical foundations.
We propose slight (but relevant) modifications of the
original definitions, which allow us to enlarge the
applicability of the GLT theory. In particular, we remove a
certain `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.
With the new definitions, we introduce new technical and
useful results and we provide a formal proof of the fact
that sequences formed by multilevel diagonal sampling
matrices, as well as multilevel Toeplitz sequences, fall in
the class of LT sequences; the latter results were
mentioned in previous papers, but no direct proof was given
especially regarding the case of multilevel diagonal
sampling matrix-sequences.
As a final step, we extend the GLT theory: 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 latter theoretical property has important
implications, e.g. in proving that the geometric means of
GLT sequences are still GLT, so obtaining for free that the
spectral distribution of the mean is just the geometric
mean of the symbols.}
}