Distribution of eigenvalues of Toeplitz matrices with smooth entries

We investigate distribution of eigenvalues of growing size Toeplitz matrices [a(n+k-j)](1 <= j,k <= n) as n -> infinity, when the entries {a(j)} are "smooth" in the sense, for example, that for some alpha > 0, a(j-1)a(j+1)/a(j)(2) = 1 - 1/a(j) (1 + o(1)), j -> infinity. Typically they are Maclaurin series coefficients of an entire function. We establish that when suitably scaled, the eigenvalue counting measures have limiting support on [0,1], and under mild additional smoothness conditions, the universal scaled and weighted limit distribution is vertical bar pi logt vertical bar(-1/2) dt on [0,1]. (C) 2021 Elsevier Inc. All rights reserved.