Asymptotic distribution of the even and odd spectra of real symmetric Toeplitz matrices (Q1970462)

Let \(T\) and \(J\) denote a given real symmetric Toeplitz matrix and the flip matrix of the same order, respectively. The eigenvectors \(e_i\) of \(T\) are known to satisfy either \(Je_i= e_i\) or \(Je_i= -e_i\), so that the spectrum of \(T\) may be split into the even spectrum comprising the eigenvalues \(\lambda_i\) whose eigenvectors satisfy \(Je_i=e_i\) and the odd spectrum comprising the other eigenvalues. In this paper, the author presents the results that describe the asymptotic distribution of both parts of the spectrum of the matrices \(T_n\) of order \(n\) with entries \(t_{ij}= {1\over\pi} \int^\pi_0 f(x)\cos (i-j)x dx\) where \(f\in L^2 [0,\pi]\) is a given even function.