On the extreme spectral properties of Toeplitz matrices generated by L(1) functions with several minima maxima

In this paper we are concerned with the asymptotic behavior of the smallest eigenvalue lambda(1)((n)) of symmetric (Hermitian) n x n Toeplitz matrices T-n(f) generated by an integrable function f defined in [-pi,pi]. In [7, 8, 11] it is shown that lambda(1)((n)) tends to 1 mf in the following way: lambda(1)((n)) - m(f) similar to 1/n(2k). These authors use three assumptions: A1) f - m(f) has a zero in x = x(0) to of order 2k. A2) f is continuous and at least C-2k in a neighborhood of x(0). A3) x = x(0) is the unique global minimum of f in [-pi,pi]. In [10] we have proved that the hypothesis of smoothness A2 is not necessary and that the same result holds under the weaker assumption that f is an element of L(1)[- pi,pi]. In this paper we further extend this theory to the case of a function f is an element of L(1)[-pi,pi] having several global minima by suppressing the hypothesis A3 and by showing that the maximal order 2k of the zeros of f - m(f) is the only parameter which characterizes the rate of convergence of lambda(1)((n)) to m(f).