On the distribution of the singular values of Toeplitz matrices

In 1920, G. Szego proved a basic result concerning the distribution of the eigenvalues {λ(n)j} of the Toeplitz sections Tn [f], where f(Θ)∈L∞( -π, π) is a real-valued function. Simple examples show that this result cannot hold in the case where f(Θ) is not real valued. In this note, we give an extension of this theorem for the singular values of Tn[f] when f(Θ)=f0(Θ)R0(Θ) with f0(Θ) real-valued and R0(Θ) continuous, periodic (with period 2π) and such that |R0(Θ)|=1. In addition, we apply the basic theorem of Szego to resolve a question of C. Moler.