Let f be a nonnegative integrable function on [ − π , π ] , T n ( f ) the ( n + 1 ) × ( n + 1 ) Toeplitz matrix associated with f and λ 1 , n its smallest eigenvalue. It is shown that the convergence of λ 1 , n to min f ( 0 ) can be exponentially fast even when f does not satisfy the smoothness condition of Kac, Murdoch and Szegö (1953). Also a lower bound for λ 1 , n corresponding to a large class of functions which do not satisfy this smoothness condition is provided.