Universal bounds on the convergence rate of extreme Toeplitz eigenvalues

Abstract Let { T n } denote the sequence of Toeplitz matrices associated with f , a non-negative integrable function such that inf f =0 and sup f >0. It is well known that T n is ill conditioned since λ min ( T n ), the smallest eigenvalue of T n , tends to zero as n →∞. If f satisfies some smoothness conditions, then the convergence rate depends on the zeros of f . Here we prove that λ min ( T n ) mimics the zeros of f only up to exponential convergence, i.e., λ min ( T n ) is always bounded from below by exp(− cn ), where c >0 depends on f , under no smoothness assumption on f . Furthermore, for multivariate f , an even stronger bound is valid. We also investigate Toeplitz matrices generated by positive measures, not necessarily absolutely continuous with respect to the Lebesgue measure, showing that in this case the convergence to zero of λ min ( T n ) can be arbitrarily fast.