The Theory of Generalized Locally Toeplitz Sequences: a Review, an Extension, and a Few Representative Applications

Abstract

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 {A n } n is a sequence of matrices with increasing size equipped with a function κ (the so-called symbol). We write {A n } n ~glt κ to indicate that {A n } n is a GLT sequence with symbol κ. This symbol characterizes the asymptotic singular value distribution of {A n } n ; if 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 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 {A n (i) } n ~glt κ i for i = 1, … ,r, and let A n = ops(A n (1) , … , A n (r) ) be a matrix obtained from A n (1) , … , A n (r) by means of certain algebraic operations “ops”, such as linear combinations, products, inversions and conjugate transpositions; then {A n } n ~glt k = ops(k 1, … , k r ).