Generalized Locally Toeplitz sequences: a review and an extension

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−1 n }n is a GLT sequence with symbol κ−1 whenever {An}n is a GLT sequence of invertible matrices with symbol κ and κ 6= 0 almost everywhere; finally, using again the approximation result, we prove that {f(An)}n is a GLT sequence with symbol f(κ), as long as f : R → R is continuous and {An}n is a GLT sequence of Hermitian matrices with symbol κ. 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.