Dichotomies for Band Matrices
Dichotomies for Band Matrices
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DOI：
10.1137/0717074
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发表时间：
1980-12
影响因子：
2.9
通讯作者：
C. Boor
中科院分区：
文献类型：
--
作者：
C. Boor
一个有界的、严格m带状的双无限矩阵a的有界可逆性（如$l_\infty $上的线性映射）被证明等价于它的核$\mathcal {N}$（如$\mathbb{R}^\mathbb{Z} $上的映射）的二分法或分裂为$\mathcal {N}^ + $和$\mathcal {N}^ - $，其中$\mathcal {N}^ + $包含在$ + \infty $指数衰减的那些，以及$\mathcal {N}^ - $在$ - \infty $指数衰减的那些。加上这个直接和分解的一定的一致性（相对于序列索引）。双无穷系统$A{\bf x} = {\bf b}$的解被该系统有限部分的解近似为线性无关，一致地表示为$I^ * \to - ( * \infty )$ ($\mathcal {N}$ / $I^ + \cup I^ - $)，其$I^ * $是一个长度为$\mathcal {N}^ * , * = + , - $的整数区间。
The bounded invertibility (as a linear map on $l_\infty $, say) of a bounded, strictly m-banded biinfinite matrix A is shown to be equivalent to a dichotomy or splitting of its kernel $\mathcal {N}$ (as a map on $\mathbb{R}^\mathbb{Z} $) into $\mathcal {N}^ + $ and $\mathcal {N}^ - $, with ,$\mathcal {N}^ + $ containing those which decay exponentially at $ + \infty $, and $\mathcal {N}^ - $ those which decay exponentially at $ - \infty $, together with a certain uniformity (with respect to the sequence index) of this direct sum decomposition. The approximability of the solution of the biinfinite system $A{\bf x} = {\bf b}$ by solutions of finite sections of this system is characterized in terms of linear independence, uniform as $I^ * \to - ( * \infty )$ of $\mathcal {N}$ over $I^ + \cup I^ - $, with $I^ * $ an integer interval of length $\mathcal {N}^ * , * = + , - $.