On the extreme eigenvalues of Toeplitz matrices

Abstract

The matrix T. [f ]=(Csj), s, j = 0,1, * * , n is called the nth finite section of the infinite Toeplitz matrix (Cs_j) associated with the function f(0). We will be concerned with functionsf(0) satisfying CONDITION A. Let f(0) be real, continuous and periodic with period 27r. Let min f(0) =f(O) = 0 and let 0 = 0 be the only value of 0 (mod 27r) for which this minimum is attained. CONDITION A(a). Let a be a positive real number. Let k(a.) be the smallest integer >oi/2. Let f(0) satisfy Condition A. Let g(0) = [f(0) ]2k/a. Let g(0) have 2k continuous derivatives in some neighborhood of 0 = 0. Finally, let g(2k)(0) -o2 >0 be the first nonvanishing derivative of g(0) at 0=0. NOTE. The a of this work is twice the a we used in [5]. We remark that the conditions f(O) = 0 and min f() =f(O) are not essential; see [2; 3 or 8]. The case o = 2 was studied by Kac, Murdock and Szego [3] who also studied a related problem for integral operators. Widom [8] also studied the case a = 2 for both the Toeplitz matrices and the related integral operators. The case a= 4 was studied by us in [4]. In [9] Widom obtained results in the integral operator case for 0 < ?2. Moreover, Widom made a correct conjecture for the case of general a in that case. The validity of his general conjecture is proven in [10]. In [5] we studied the Toeplitz matrices for the case where a is an eveni integer. We showed that if X,n are the eigenvalues of T. [f ] arranged in nondecreasing order, then for fixed v as noo we have