@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.}
}