In a sequence of previous works with Albrecht Bottcher, 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 W-alpha for alpha >= 4, or satisfies even stronger smoothness conditions. We now use a different technique, which allows us to extend previous results to the case alpha >= 1 with additional smoothness at the minimum and maximum points.

Исследование асимптотического поведения спектральных характеристик тeплицевых матриц, когда размерность матрицы стремится к бесконечности, имеет более чем столетнюю историю. Например, хорошо известны многочисленные варианты теоремы Сeгe об асимптотическом распределении собственных чисел и так называемой сильной теоремы Сегe об асимптотическом поведении определителей тeплицевых матриц. Начиная с 1950-х гг. интенсивно изучались асимптотики наибольших и наименьших собственных чисел. Отметим однако, что исследования, посвященные изучению индивидуальных асимптотик всех собственных чисел и собственных векторов матриц Тeплица, появились совсем недавно. Первые статьи на эту тему опубликованы в 2009-2010 гг. Настоящая работа посвящена обзору этого нового направления. Библиография: 55 названий.

We present conditions that allow us to pass from the convergence of probability measures in distribution to the uniform convergence of the associated quantile functions. Under these conditions, one can in particular pass from the asymptotic distribution of collections of real numbers, such as the eigenvalues of a family of n-by-n matrices as n goes to infinity, to their uniform approximation by the values of the quantile function at equidistant points. For Hermitian Toeplitz-like matrices, convergence in distribution is ensured by theorems of the Szegő type. Our results transfer these convergence theorems into uniform convergence statements.

The collective behavior of the singular values of large Toeplitz matrices is described by the Avram–Parter theorem. In the case of Hermitian matrices, the Avram–Parter theorem is equivalent to Szegő’s theorem on the eigenvalues. The Avram–Parter theorem in conjunction with an improvement made by Trench implies estimates in the mean between the singular values and the appropriately ordered absolute values of the symbol. The purpose of this paper is twofold. Under natural hypotheses, we first strengthen the known estimates in the mean to estimates in the maximum norm, thus turning from collective results on the singular values to results on individual singular values. Secondly, we want to emphasize that the use of the quantile function eases the proofs and statements of results significantly and provides a promising language for forthcoming research into higher order asymptotics for individual singular values.

The paper presents higher-order asymptotic formulas for the eigenvalues of large Hermitian Toeplitz matrices with moderately smooth symbols which trace out a simple loop on the real line. The formulas are established not only for the extreme eigenvalues, but also for the inner eigenvalues. The results extend and make more precise existing results, which so far pertain to banded matrices or to matrices with infinitely differentiable symbols. Also given is a fixed-point equation for the eigenvalues which may be solved numerically by an iteration method.