How bad can positive definite Toeplitz matrices be?

Let f is an element of L-1(-pi, pi), f greater than or equal to 0, and suppose f does not vanish identically. Denote by T-n(f) the n x n Toeplitz matrix generated by f and let kappa(T-n(f)) stand for the spectral condition number of T-n(f). Recently we proved that if f is not pathological, then kappa(T-n(f))= O(e(gamma n2)) for some constant gamma > O. In this note we show that an approximation theorem by Yano easily implies the sharper estimate kappa(T-n(f)) = O(e(gamma n)). We also characterize a class of functions f for which kappa(T-n(f)) similar or equal to e(gamma n), and we establish a conjecture concerning pathological functions f.