论文标题
希尔伯特空间中环状投影方法的多项式估计值
Polynomial Estimates for the Method of Cyclic Projections in Hilbert Spaces
论文作者
论文摘要
当应用于封闭式和线性子空间$ m_i $，$ i = 1，\ ldots，m $，是真正的Hilbert Space $ \ Mathcal H $时，我们研究了循环预测的方法。我们表明，与单个集合的平均距离沿着生成的迭代的轨迹享受多项式行为$ O（k^{ - 1/2}）$。令人惊讶的是，当从子空间$ \ sum_ {i = 1}^{m} m_i^\ perp $选择起点时，我们的结果会产生收敛的多项式速率$ \ MATHCAL O（k^{ - 1/2}）$用于循环预测本身的方法。此外，如果$ \ sum_ {i = 1}^{m} m_i^\ perp $未关闭，那么上述两个速率最好从某种意义上说，从相应的多项式$ k^{1/2} $无法替换为$ k^{1/2+\ varepsilon} $ for $ $ k^{1/2} $ for $ k^{
We study the method of cyclic projections when applied to closed and linear subspaces $M_i$, $i=1,\ldots,m$, of a real Hilbert space $\mathcal H$. We show that the average distance to individual sets enjoys a polynomial behaviour $o(k^{-1/2})$ along the trajectory of the generated iterates. Surprisingly, when the starting points are chosen from the subspace $\sum_{i=1}^{m}M_i^\perp$, our result yields a polynomial rate of convergence $\mathcal O(k^{-1/2})$ for the method of cyclic projections itself. Moreover, if $\sum_{i=1}^{m} M_i^\perp$ is not closed, then both of the aforementioned rates are best possible in the sense that the corresponding polynomial $k^{1/2}$ cannot be replaced by $k^{1/2+\varepsilon}$ for any $\varepsilon >0$.
