On the extreme eigenvalues of Hermitian (block) Toeplitz matrices

We are concerned with the behavior of the minimum (maximum) eigenvalue lambda(0)((n)) (lambda(n)((n))) of an (n + 1)x(n + 1) Hermitian Toeplitz matrix T-n(f) where f is an integrable real-valued function. Kac, Murdoch, and Szego, Widom, Parter, and R. H. Chan obtained that lambda(0)((n)) - min f = O(1/n(2k)) in the case where f is an element of C-2k at least locally, and f - inf f has a zero of order 2k. We obtain the same result under the second hypothesis alone. Moreover we develop a new tool in order to estimate the extreme eigenvalues of the mentioned matrices, proving that the rate of convergence of lambda(0)((n)) to inf f depends only on the order rho (not necessarily even or integer or finite) of the zero of f - inf f. With the help of this tool, we derive an absolute lower bound for the minimal eigenvalues of Toeplitz matrices generated by nonnegative L-1 functions and also an upper bound for the associated Euclidean condition numbers. Finally, these results are extended to the case of Hermitian block Toeplitz matrices with Toeplitz blocks generated by a bivariate integrable function f. (C) 1998 Elsevier Science Inc.