Asymptotic dependence of the extreme eigenvalues of truncated Toeplitz matrices on the rate at which the symbol attains its extremum

In this paper we investigate the asymptotic behavior of extreme eigenvalues for truncated Toeplitz (N × N)-matrices TN with real-valued symbol a ∈ L∞(S) such that the function a(t) − inft∈S a(t) has finitely many zeros on the unit circle S of the complex plane, and the maximal order of a zero equals ν > 0. Our main results are Theorems 2.1 and 2.2 saying that, as N → +∞, the lowest eigenvalues of the matrices TN tend to inft∈S a(t) with the rate 1/N . Our research can be viewed as a response to the paper [1], where a similar result was obtained only for even ν, and in some cases estimates were obtained in terms of the even numbers neighboring with ν. Those results were based on the fundamental results of [2, 3], which can also be found in the monograph [4]. Our method does not employ those results, and it allows us to obtain upper and lower estimates of sharp order for any ν > 0.