Let $L_1,L_2,\dots ,L_K$ be a family of closed subspaces of a Hilbert space $H$, $L_1\cap \cdots \cap L_K=\left\{0\right\}$; let $P_k$ be the orthogonal projection onto $L_k$. We consider two types of consecutive projections of an element $x_0\in H$: alternating projections $T^n{x}_0$, where $T=P_K\circ \cdots \circ P_1$, and remotest projections $x_n$ defined recursively, $x_{n+1}$ being the remotest point for $x_n$ among $P_1{x}_n,\dots ,P_K{x}_n$. These $x_n$ can be interpreted as residuals in greedy approximation with respect to a special dictionary associated with $L_1,L_2,\dots ,L_K$. We establish parallels between convergence properties separately known for alternating projections, remotest projections, and greedy approximation in $H$. Here are some results. If $L_1^{\perp }+\cdots +L_K^{\perp }=H$, then $x_n\to 0$ exponentially fast. In case $L_1^{\perp }+\cdots +L_K^{\perp }\ne H$, the convergence $x_n\to 0$ can be arbitrarily slow for certain $x_0$. Such a dichotomy, exponential rate of convergence everywhere on $H$, or arbitrarily slow convergence for certain starting elements, is valid for greedy approximation with respect to general dictionaries. The dichotomy was known for alternating projections. Using the methods developed for greedy approximation we prove that $\left|T^n{x}_0\right|\le C\left(x_0\right)n^{-\alpha \left(K\right)}$ for certain positive $\alpha \left(K\right)$ and all starting points $x_0\in L_1^{\perp }+\cdots +L_K^{\perp }$.

让 ${\mathrm{大号}}_{\mathrm{1个}}，{\mathrm{大号}}_2，\dots ，{\mathrm{大号}}_{ķ}$ 是希尔伯特空间的封闭子空间的族 $H$， ${\mathrm{大号}}_{\mathrm{1个}}\cap \cdots \cap {\mathrm{大号}}_{ķ}=\left\{0\right\}$; 让$P_{ķ}$ 正交投影到 ${\mathrm{大号}}_{ķ}$。我们考虑元素的两种连续投影$X_0\in H$：交替投影 ${Ť}^{ñ}{X}_0$，在哪里 $Ť=P_{ķ}\circ \cdots \circ P_{\mathrm{1个}}$和最远的投影 $X_{ñ}$ 递归定义 $X_{ñ+\mathrm{1个}}$ 是最遥远的地方 $X_{ñ}$ 其中 $P_{\mathrm{1个}}{X}_{ñ}，\dots ，P_{ķ}{X}_{ñ}$。这些$X_{ñ}$ 可以解释为相对于与 ${\mathrm{大号}}_{\mathrm{1个}}，{\mathrm{大号}}_2，\dots ，{\mathrm{大号}}_{ķ}$。我们在收敛性之间建立了平行关系，分别针对交替投影，最远投影和贪婪近似$H$。这是一些结果。如果${\mathrm{大号}}_{\mathrm{1个}}^{\perp }+\cdots +{\mathrm{大号}}_{ķ}^{\perp }=H$， 然后 $X_{ñ}\to 0$指数级快速。以防万一${\mathrm{大号}}_{\mathrm{1个}}^{\perp }+\cdots +{\mathrm{大号}}_{ķ}^{\perp }\ne H$，收敛 $X_{ñ}\to 0$ 可以在一定程度上任意慢 $X_0$。这种二分法，无处不在的指数收敛速度$H$，或某些起始元素的任意慢收敛，对于一般词典而言，对于贪婪近似有效。二分法以交替投影而闻名。使用为贪婪近似开发的方法，我们证明了$\left|{Ť}^{ñ}{X}_0\right|\le C（X_0）{ñ}^{-\alpha （ķ）}$ 对于某些肯定 $\alpha （ķ）$ 和所有起点 $X_0\in {\mathrm{大号}}_{\mathrm{1个}}^{\perp }+\cdots +{\mathrm{大号}}_{ķ}^{\perp }$。