REMARKS ON EXTREME EIGENVALUES OF TOEPLITZ MATRICES

ABSTRACT. Let f be a nonnegative integrable function on [-r,r], T.Cf) the Ca+l) X(n+l) Toeplitz matrix associated with f and k,. its smallest eigenvalue. It is shown that the convergence of)t,. to rain f(O) can be exrmnentiallv fast even when f does not satisfy the smoothness condition of Kae, Murdoeh and Szeg6 (1953). Also a lower bound for),,. corresponding to a large class of functions which do not satisfy this smoothness condition is provided. KEY WORDS AND PHRASES. Toeplitz matrices, smallest eigenvalue, stationary process