Singular values and eigenvalues of non-Hermitian block Toeplitz matrices

The asymptotic distribution of singular values and eigenvalues of non-Hermitian block Toeplitz matrices is studied. These matrices are associated with the Fourier series of an univariate function f. The asymptotic distribution of singular values is computed when f belongs to L-2 and is matrix-valued, not necessarily square. Clusters of singular values are also studied, and a new result is proved. Moreover, a classical formula due to Szego concerning the asymptotic spectrum of Hermitian Toeplitz matrices is extended to the non-Hermitian block case, under the assumption that f is bounded and test functions are harmonic. Finally, it is proved that the class of harmonic test functions is optimal, as far as that formula is concerned. (C) 1998 Elsevier Science Inc.