Remarks on extreme eigenvalues of Toeplitz matrices

Abstract

Let f be a nonnegative integrable function on [−π,π], Tn(f) the (n+1)×(n+1) Toeplitz matrix associated with f and λ1,n its smallest eigenvalue. It is shown that the convergence of λ1,n to minf(0) can be exponentially fast even when f does not satisfy the smoothness condition of Kac, Murdoch and Szegö (1953). Also a lower bound for λ1,n corresponding to a large class of functions which do not satisfy this smoothness condition is provided.