On the extreme spectral properties of toeplitz matrices generated by L1 functions with several minima/maxima

In this paper we are concerned with the asymptotic behavior of the smallest eigenvalue λ1(n) of symmetric (Hermitian) n × nToeplitz matrices Tn(f) generated by an integrable function f defined in [-π, π]. In [7, 8, 11] it is shown that λ1(n) tends to essinf f = mf in the following way: λ1(n) - mf ∼ 1/n2k. These authors use three assumptions: A1) f - mf has a zero in x = x0 of order 2k. A2) f is continuous and at least C2k in a neighborhood of x0. A3) x = x0 is the unique global minimum of f in [-π, π]. 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 ∈ L1[-π,π]. In this paper we further extend this theory to the case of a function f ∈ L1 [-π, π] having several global minima by suppressing the hypothesis A3 and by showing that the maximal order 2k of the zeros of f - mf is the only parameter which characterizes the rate of convergence of λ1(n) to m