LOCALLY TOEPLITZ SEQUENCES - SPECTRAL PROPERTIES AND APPLICATIONS

The notion of locally Toeplitz sequence of matrices is introduced, whi ch extends the notion of Toeplitz sequence of matrices. The singular V alue distribution and, in the Hermitian case, the eigenvalue distribut ion is completely characterized for locally Toeplitz sequences and the ir sums, obtaining weighted Szego formulas which extend well-known res ults, due to Tyrtyshnikov, concerning Toeplitz matrices; indeed, any T oeplitz sequence {T-n(f)}, where f is bounded, is proved to be a local ly Toeplitz sequence. Moreover, sufficient conditions are given for th e product of two locally Toeplitz sequences to be also locally Toeplit z. By combining these theoretic results, we are able to explicitly com pute the asymptotic spectral distribution of a large class of matrices arising in the applications, including the algebra generated by Toepl itz sequences, and virtually all matrices resulting from the discretiz ation of a unidimensional differential operator with non-constant coef ficients. Finally, a large number of examples is discussed. (C) 1998 E lsevier Science Inc. All rights reserved.