The theory of generalized Locally Toeplitz sequences: A review, an extension, and a few representative applications

We review and extend the theory of Generalized Locally Toeplitz (GLT) sequences, which goes back to Tilli’s work on Locally Toeplitz sequences and was developed by the second author during the last decade. Informally speaking, a GLT sequence Ann is a sequence of matrices with increasing size equipped with a function κ (the so-called symbol). We write Ann ∼GLT κ to indicate that Ann is a GLT sequence with symbol κ. This symbol characterizes the asymptotic singular value distribution of Ann; if the matrices An are Hermitian, it also characterizes the asymptotic eigenvalue distribution of Ann. Three fundamental examples of GLT sequences are: (i) the sequence of Toeplitz matrices generated by a function f in L1; (ii) the sequence of diagonal sampling matrices containing the samples of a Riemann-integrable function a over equispaced grids; (iii) any zero-distributed sequence, i.e., any sequence of matrices with an asymptotic singular value distribution characterized by 0. The symbol of the GLT sequence (i) is f, the symbol of the GLT sequence (ii) is a, and the symbol of the GLT sequences (iii) is 0. The set of GLT sequences is a *-algebra. More precisely, suppose that An(i) n ∼GLT κi for i = 1,…, r, and let An = ops(An(1) ,…, An(r) be a matrix obtained from An(1) ,…, An(r) by means of certain algebraic operations “ops”, such as linear combinations, products, inversions and conjugate transpositions; then Ann ∼GLT κ = ops(κ1,…, κr). The theory of GLT sequences is a powerful apparatus for computing the asymptotic singular value and eigenvalue distribution of the discretization ma-trices An 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 matrices An give rise to a sequence Ann, which often turns out to be a GLT sequence. Nevertheless, this work is not primarily concerned with the applicative interest of the theory of GLT sequences. Although we will provide some illustrative applications at the end, the attention is focused on the mathematical foundations of the theory. We first propose a modification of the original definition of GLT sequences. With the new definition, we are able to enlarge the applicability of the theory, by generalizing/simplifying a lot of key results. In particular, we remove the Riemann-integrability assumption from the main spectral distribution and algebraic results for GLT sequences. As a final step, we extend the theory. We first prove an approximation result, which is 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 An-1 n ∼GLT κ−1 whenever Ann ∼GLT κ, the matrices An are invertible, and κ ≠ 0 almost everywhere. Finally, using again the approximation result, we prove that f(An)n ∼GLT f(κ) whenever Ann ∼GLT κ, the matrices An are Hermitian, and f : R → R is continuous.