关于Toeplitz矩阵计算的四个短篇故事
Four short stories about Toeplitz matrix calculations
AI总结：
本文通过四个故事研究有限、无限和双无限Toeplitz系统的求解，利用Banach代数理论刻画矩阵衰减，推导有限截面法和循环嵌入的误差估计，为反卷积数值现象提供理论解释。
摘要：
本文所述的故事涉及有限、无限和双无限Toeplitz型系统的求解。Toeplitz矩阵及其逆矩阵的非对角衰减行为起着关键作用。Gelfand等人关于交换Banach代数的经典结果给出了这种衰减行为的一般刻画。然后，我们推导了通过有限截面法近似求解（双）无限Toeplitz系统的估计，表明近似速率仅取决于Toeplitz矩阵元素的衰减及其条件数。此外，我们给出了通过有限循环系统求解双无限卷积系统的误差估计。最后，推导了通过循环嵌入构造预条件子的一些定量结果，这为一些研究人员在反卷积问题中观察到的数值现象提供了理论解释。
英文摘要：
The stories told in this paper are dealing with the solution of finite, infinite, and biinfinite Toeplitz-type systems. A crucial role plays the off-diagonal decay behavior of Toeplitz matrices and their inverses. Classical results of Gelfand et al. on commutative Banach algebras yield a general characterization of this decay behavior. We then derive estimates for the approximate solution of (bi)infinite Toeplitz systems by the finite section method, showing that the approximation rate depends only on the decay of the entries of the Toeplitz matrix and its condition number. Furthermore, we give error estimates for the solution of doubly infinite convolution systems by finite circulant systems. Finally, some quantitative results on the construction of preconditioners via circulant embedding are derived, which allow to provide a theoretical explanation for numerical observations made by some researchers in connection with deconvolution problems.
机构、总结、中文摘要均由AI分析而来，可能存在错误，请以论文正文为准。
