On the extreme eigenvalues of Toeplitz matrices
Seymour V. Parter · Transactions of the American Mathematical Society · 1961
Introduction. Beginning with work of Kac, Murdock, and Szegö [3] (see [2] also) several recent works have been concerned with the asymptotic behavior of the extreme eigenvalues of Toeplitz matrices.Let f(d) be a real valued, Lebesgue integrable function defined on [-ir, ir].Let { Cj] be the Fourier coefficients of f(d), i.e.,The matrix Tn[/] = (C"_y), 5, 7 = 0, 1, •• -, re is called the wth finite section of the infinite Toeplitz matrix (C,-¡) associated with the function f(9).We will be concerned with functions f(6) satisfying Condition A. Let f(d) be real, continuous and periodic with period 2w.Let min f(B) =/(0) = 0 and let 6 = 0 be the only value of 0 (mod 2tt) for which this minimum is attained.Condition A(a).Let a be a positive real number.Let k(a) be the smallest integer £a/2.Letf(6) satisfy Condition A. Let g(6) = [f(0) ]2*/a.Let g(0) have 2k continuous derivatives in some neighborhood of 0 = 0. Finally, let g(2*}(0) = 0 be the first nonvanishing derivative of g(0) at 6 = 0. Note.The a of this work is twice the a we used in [5].We remark that the conditions/(0) =0 and min/(0) =/(0) are not essen-