An Elementary Note on Asymptotic Properties of Toeplitz and Multilevel Toeplitz Matrices

Let be the Fourier coefficients of ∈ [−, ] and consider the Toeplitz matrices {()}, where ; thus, {()} is generated by the symbol . There are many results on asymptotic properties of {()} as → ∞. If is a positive integer and , then where is easily obtained from . Hence, known results on the asymptotic behavior of Toeplitz matrices generated by a symbol can be applied to . Although this is elementary, to our knowledge it has not been previously exploited. We extend this idea to multilevel Toeplitz matrices.15A18; 15A57