Eigenvalues of hermitian toeplitz matrices generated by simple-loop symbols with relaxed smoothness

In a sequence of previous works with Albrecht Böttcher, we established higher-order uniform individual asymptotic formulas for the eigenvalues and eigenvectors of large Hermitian Toeplitz matrices generated by symbols satisfying the so-called simple-loop condition, which means that the symbol has only two intervals of monotonicity, its first derivative does not vanish on these intervals, and the second derivative is different from zero at the minimum and maximum points. Moreover, in previous works it was supposed that the symbol belongs to the weighted Wiener algebra Wα for α ≥ 4, or satisfies even stronger smoothness conditions. We now use a different technique, which allows us to extend previous results to the case α ≥ 1 with additional smoothness at the minimum and maximum points.