Asymptotic distribution of the even and odd spectra of real symmetric Toeplitz matrices

If T-n = (t(r-s))(r,s=0)(n) is a real symmetric Toeplitz (RST) matrix then R-n has a basis consisting of [n/2] eigenvectors x satisfying (A) Jx = x and [n/2] eigenvectors y satisfying (B) Jy = -y, where is the flip matrix. We say that an eigenvalue lambda of T-n is even if a lambda-eigenvector of T-n satisfies (A), or odd if a lambda-eigenvector of T-n satisfies (B). We call the collection of even (odd) eigenvalues of T, the even (odd) spectrum of T,. In the case where t(r) = 1/pi integral(0)(pi) f(x) cos rxdx a great deal is known about the asymptotic distribution of the eigenvalues of T-n as n -> infinity, under suitable assumptions on f. However, the question of the separate asymptotic distributions of the even and odd spectra does not seem to have been raised. This is the subject of this paper. (C) 1999 Elsevier Science Inc. All rights reserved.