A unifying approach to some old and new theorems on distribution and clustering Original Research Article

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 Szegimage and Avram-Parter theorems, where the generating function is now required to belong to L2, and not necessarily to L∞. Analogous extensions are given for multilevel Toeplitz matrices. In particular, it is proved that if f(x1, …, xp)set membership, variant L2, then the p-level (complex) Toeplitz matrices allied with f have their singular values distributed as f(x1, …, xp). The distribution results for the Cesàro (optimal) circulants hold even if f set membership, variant L1. Also suggested are new theorems on clustering that have to do with the preconditioning of multilevel Toeplitz matrices by multilevel circulants.