Abstract
A unifying approach is proposed to studying the distributions of eigenvalues and singular values of Toeplitz matrices associated with a Fourier series, and multilevel Toeplitz matrices associated with a multidimensional Fourier series. Obtained are the extensions of the Szegő and Avram-Parter theorems, where the generating function is now required to belong to L 2, and not necessarily to L ∞. Analogous extensions are given for multilevel Toeplitz matrices. In particular, it is proved that if f( x 1, …, x p )∈ L 2, then the p-level (complex) Toeplitz matrices allied with f have their singular values distributed as | f( x 1, …, x p )|. The distribution results for the Cesàro (optimal) circulants hold even if f ∈ L 1. Also suggested are new theorems on clustering that have to do with the preconditioning of multilevel Toeplitz matrices by multilevel circulants.
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