Locally Toeplitz sequences: spectral properties and applications

Abstract

The notion of locally Toeplitz sequence of matrices is introduced, which extends the notion of Toeplitz sequence of matrices. The singular value distribution and, in the Hermitian case, the eigenvalue distribution is completely characterized for locally Toeplitz sequences and their sums, obtaining weighted Szegő formulas which extend well-known results, due to Tyrtyshnikov, concerning Toeplitz matrices; indeed, any Toeplitz sequence { T n ( f)}, where f is bounded, is proved to be a locally Toeplitz sequence. Moreover, sufficient conditions are given for the product of two locally Toeplitz sequences to be also locally Toeplitz. By combining these theoretic results, we are able to explicitly compute the asymptotic spectral distribution of a large class of matrices arising in the applications, including the algebra generated by Toeplitz sequences, and virtually all matrices resulting from the discretization of a unidimensional differential operator with non-constant coefficients. Finally, a large number of examples is discussed.