Approximating inverses of Toeplitz matrices by circulant matrices (Q2577079)

To a continuous complex-valued function \(A\) on the complex unit circle, one can associate a sequence \(\{T_n(a)\}_{n=1}^\infty\) of Toeplitz matrices and a sequence \(\{C_n(a)\}_{n=1}^\infty\) of circulant matrices. In the paper under review, the authors consider the problem of estimating the difference \(T_n^{-1}(a)-C_n^{-1}(a)\) in some sense. They prove asymptotic estimates for the central columns of the matrices \(T_n^{-1}(a)-C_n^{-1}(a)\) as \(n\to\infty\). Their results generalize and sharpen the recent results by \textit{T.~Strohmer} [Linear Algebra Appl.\ 343/344, 321--344 (2002; Zbl 0999.65026)] and by \textit{F.--W.\ Sun, Y.~Jiang} and \textit{J.~S.\ Baras} [IEEE Trans.\ Inf.\ Theory 49, No.~1, 180--190 (2003; Zbl 1063.15024)].