A generalization of the Avram–Parter Theorem

The First Szegő Limit Theorem gives a remarkable connection between the eigenvalues distribution of a Hermitian Toeplitz matrix and its symbol. This result was extended by Avram and Parter to the singular values of complex Toeplitz matrices. The purpose of this note is to extend their results to a larger class of matrices whose entries are equidistributed and have small mean variation. We also present applications to Kac–Murdock–Szegő matrices, block Toeplitz and locally Toeplitz matrices.