On the distribution of the singular values of Toeplitz matrices (Q1080917)

The involved discussion is based on a theorem of Szegö concerning the asymptotic distribution of the eigenvalues of the Toeplitz matrices \(T_ n[f]\) where f(0) is a real-valued bounded measurable function which is periodic with period \(2\pi\). Simple examples show that a similar theorem for the case where f(0) is not real-valued is impossible. An interlacing problem for singular values is proved. The theorem, although of a more general nature than usual, has a proof which is essentially the proof of the interlacing theorem for Hermitian matrices. The theorem is applied to obtain extensions of the \textit{G. Szegö} theorem [Math. Zeitschr. 6, 167-202 (1920; JFM 47.0391.04)] to the singular values of \(T_ n[f]\) when f is not a real-valued function.