Abstract :
Let {Tn} denote the sequence of Toeplitz matrices associated with f, a non-negative integrable function such that inff=0 and supf>0. It is well known that Tn is ill conditioned since λmin(Tn), the smallest eigenvalue of Tn, 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 λmin(Tn) mimics the zeros of f only up to exponential convergence, i.e., λmin(Tn) 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 λmin(Tn) can be arbitrarily fast.
