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.

[1]  E. E. Tyrtyshnikov.  A unifying approach to some old and new theorems on distribution and clustering , 1996 .

[2]  L. Collatz.  The numerical treatment of differential equations , 1961 .

[3]  H. Widom.  Asymptotic behavior of block Toeplitz matrices and determinants. II , 1974 .

[4]  Seymour V. Parter,et al.  On the distribution of the singular values of Toeplitz matrices , 1986 .

[5]  U. Grenander,et al.  Toeplitz Forms And Their Applications , 1958 .

[6]  F. Avram.  On bilinear forms in Gaussian random variables and Toeplitz matrices , 1988 .

[7]  P. Tilli.  Singular values and eigenvalues of non-hermitian block Toeplitz matrices , 1996 .

[8]  H. Widom.  Szegö's limit theorem: The higher-dimensional matrix case☆ , 1980 .

[9]  H. Widom.  On the limit of block Toeplitz determinants , 1975 .

[10]  Eugene E. Tyrtyshnikov,et al.  Spectra of multilevel toeplitz matrices: Advanced theory via simple matrix relationships , 1998 .

[11]  Stefano Serra,et al.  The rate of convergence of Toeplitz based PCG methods for second order nonlinear boundary value problems , 1999 .

[12]  Charles R. Johnson,et al.  Matrix analysis , 1985, Statistical Inference for Engineers and Data Scientists.

[13]  Ali H. Sayed,et al.  Displacement Structure: Theory and Applications , 1995, SIAM Rev..

[14]  W. Rudin.  Real and complex analysis , 1968 .

[15]  E. E. Tyrtyshnikov.  Influence of matrix operations on the distribution of Eigenvalues and singular values of Toeplitz matrices , 1994 .