Spectral Distribution of Hermitian Toeplitz Matrices Formally Generated by Rational Functions

Abstract. We consider the asymptotic spectral distribution of Hermitian Toeplitz matrices {Tn}∞n=1 formally generated by a rational function h(z) = (f(z)f∗(1/z))/(g(z)g∗(1/z)), where the numerator and denominator have no common zeros, deg(f) < deg(g), and the zeros of g are in the open punc-tured disk 0 < |z | < 1. From Szegö’s theorem, the eigenvalues of {Tn} are distributed like the values of h(eiθ) as n → ∞ if Tn = (tr−s)nr,s=1, where {t`} `=− ∞ are the coefficients in the Laurent series for h that converges in an annulus containing the unit circle. We show that if {t`} `=− ∞ are the coeffi-cients in certain other formal Laurent series for h, then there is an integer p such that all but the p smallest and p largest eigenvalues of Tn are distributed like the values of h(eiθ) as n→∞