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On the distribution of the singular values of Toeplitz matrices. (English) Zbl 0601.15006

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 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.
Reviewer: M.de la Sen

MSC:

15A18 Eigenvalues, singular values, and eigenvectors
15B57 Hermitian, skew-Hermitian, and related matrices
15A23 Factorization of matrices

Citations:

JFM 47.0391.04

References:

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