On the extreme spectral properties of Toeplitz matrices generated byL1 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 ×n Toeplitz matricesTn(f) generated by an integrable functionf defined in [−π, π]. In [7, 8, 11] it is shown that λ1(n) tends to essinff =mf in the following way: λ1(n) −mf ∼ 1/n2k. These authors use three assumptions:A1)f −mf has a zero inx =x0 of order 2k.A2)f is continuous and at leastC2k in a neighborhood ofx0.A3)x =x0 is the unique global minimum off in [−π, π]. In [10] we have proved that the hypothesis of smoothnessA2 is not necessary and that the same result holds under the weaker assumption thatf εL1[−π, π]. In this paper we further extend this theory to the case of a functionf εL1[−π, π] having several global minima by suppressing the hypothesisA3 and by showing that the maximal order 2k of the zeros off −mf is the only parameter which characterizes the rate of convergence of λ1(n) tomf.