Approximating Inverse of Toeplitz Matrices by Circulant Matrices

Abstract

With every continuous function a on the complex unit circle one can associate a sequence {T_n(a)}^infinity _n=1 of Toeplitz matrices and a sequence {C_n(a)} ^infinity_n=1 of circulant matrices. By employing some advanced results on the finite sections of Toeplitz operators, we prove asymptotic estimates for the central columns of the matrices T_n^-1(a) - C_n ^-1(a)as n goes to infinity. Our results generalize and sharpen recent results by T. Strohmer and by F.-W. Sun, Y. Jiang, and J. S. Baras, who also discussed the relevancy of the problem in signal processing.