Asymptotics of eigenvalues and eigenvectors of Toeplitz matrices

Analysis of the asymptotic behavior of the spectral characteristics of Toeplitz matrices as the dimension of the matrix tends to infinity has a history of over 100 years. For instance, quite a number of versions of Szegö`s theorem on the asymptotic behavior of eigenvalues and of the so-called strong Szegö theorem on the asymptotic behavior of the determinants of Toeplitz matrices are known. Starting in the 1950s, the asymptotics of the maximum and minimum eigenvalues were actively investigated. However, investigation of the individual asymptotics of all the eigenvalues and eigenvectors of Toeplitz matrices started only quite recently: the first papers on this subject were published in 2009-2010. A survey of this new field is presented here.