Distribution of eigenvalues of Toeplitz matrices with smooth entries

We investigate distribution of eigenvalues of growing size Toeplitz matrices [an+k−j]1≤j,k≤n as n→∞, when the entries {aj} are “smooth” in the sense, for example, that for some α>0,aj−1aj+1aj2=1−1αj(1+o(1)),j→∞. 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 |πlog⁡t|−1/2dt on [0,1].