Approximating inverses of Toeplitz matrices by circulant matrices. (English) Zbl 1097.47028
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 T. Strohmer [Linear Algebra Appl.343/344, 321–344 (2002; Zbl 0999.65026)] and by F.–W.Sun, Y. Jiang and J. S.Baras [IEEE Trans.Inf.Theory 49, No. 1, 180–190 (2003; Zbl 1063.15024)].
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 T. Strohmer [Linear Algebra Appl.343/344, 321–344 (2002; Zbl 0999.65026)] and by F.–W.Sun, Y. Jiang and J. S.Baras [IEEE Trans.Inf.Theory 49, No. 1, 180–190 (2003; Zbl 1063.15024)].
Reviewer: Woo Young Lee (Seoul)
MSC:
| 47B35 | Toeplitz operators, Hankel operators, Wiener-Hopf operators |
| 15A60 | Norms of matrices, numerical range, applications of functional analysis to matrix theory |
| 65F10 | Iterative numerical methods for linear systems |
| 47N70 | Applications of operator theory in systems, signals, circuits, and control theory |
| 94A12 | Signal theory (characterization, reconstruction, filtering, etc.) |
