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

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 Szego and Avram-Parter theorems, where the generating function is now required to belong to L(2), and not necessarily to L(infinity). Analogous extensions are given for multilevel Toeplitz matrices. In particular, it is proved that if f(x(1),...,x(p)) is an element of 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 Cesaro (optimal) circulants hold even if f is an element of L(1). Also suggested are new theorems on clustering that have to do with the preconditioning of multilevel Toeplitz matrices by multilevel circulants.