循环交替投影法第一步的最佳误差界
On the optimal error bound for the first step in the method of cyclic alternating projections
摘要：
设$H$是Hilbert空间，$H_1,...,H_n$是$H$的闭子空间。设置$H_0:=H_1\cap H_2\cap...\cap H_n$，并使$P_k$为$H_k$、$k=0,1,...,n$上的正交投影。本文研究了由$$ f_n(c)=\sup\{\|P_n...P_2 P_1-P_0\|\,|c_F(H_1,...,H_n)\leqslant c\},\,c\in[0,1], $$定义的函数$f_n:[0,1]\to\mathbb{R}$，其中上确界是所有Friedrichs数$c_F(H_1,...,H_n)$小于或等于$c$的子空间$H_1,...,H_n$的系统的上确界.利用$f_n$函数可以很容易地得到循环交替投影法收敛速度的上界。我们将证明，问题的$f_n(c)$是等价于一定的优化问题的一个子集的厄米特复杂的$n\times n$矩阵。利用等价性我们找到了$f_3$，并研究了$f_n$，$n\geqslant 4$的性质.此外，我们证明了对于所有的$c\in[0,1]$，其中$a_n=2(n-1)\sin^2(π/(2n))$，$b_n=6(n-1)^2\sin^4(π/(2n))$和$\widetilde{b}_n$是某个正数，$$ 1-a_n(1-c)-\widetilde{b}_n(1-c)^2\leqslant f_n(c)\leqslant 1-a_n(1-c)+b_n(1-c)^2 $$。
英文摘要：
Let $H$ be a Hilbert space and $H_1,...,H_n$ be closed subspaces of $H$. Set $H_0:=H_1\cap H_2\cap...\cap H_n$ and let $P_k$ be the orthogonal projection onto $H_k$, $k=0,1,...,n$. The paper is devoted to the study of functions $f_n:[0,1]\to\mathbb{R}$ defined by $$ f_n(c)=\sup\{\|P_n...P_2 P_1-P_0\|\,|c_F(H_1,...,H_n)\leqslant c\},\,c\in[0,1], $$ where the supremum is taken over all systems of subspaces $H_1,...,H_n$ for which the Friedrichs number $c_F(H_1,...,H_n)$ is less than or equal to $c$. Using the functions $f_n$ one can easily get an upper bound for the rate of convergence in the method of cyclic alternating projections. We will show that the problem of finding $f_n(c)$ is equivalent to a certain optimization problem on a subset of the set of Hermitian complex $n\times n$ matrices. Using the equivalence we find $f_3$ and study properties of $f_n$, $n\geqslant 4$. Moreover, we show that $$ 1-a_n(1-c)-\widetilde{b}_n(1-c)^2\leqslant f_n(c)\leqslant 1-a_n(1-c)+b_n(1-c)^2 $$ for all $c\in[0,1]$, where $a_n=2(n-1)\sin^2(π/(2n))$, $b_n=6(n-1)^2\sin^4(π/(2n))$ and $\widetilde{b}_n$ is some positive number.
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