论文标题
在逆最佳近似属性上，希尔伯特空间的子空间的系统
On the inverse best approximation property of systems of subspaces of a Hilbert space
论文作者
论文摘要
令$ h $为希尔伯特空间，$ h_1，...，h_n $是$ h $的子空间。用$ p_k $表示正交投影到$ h_k $，$ k = 1,2，...，n $。遵循Patrick L. Combettes和Noli N. Reyes，我们会说子空间的系统$ h_1，...，h_n $具有相反的最佳近似属性（IBAP），如果是任意元素$ x_1 \ in H_1，...，...，...，x_n $ in h_n $ in h_n $ in h_n $都存在$ x $ x $ x $ x $ x $ $ k = 1,2，...，n $。我们为拥有IBAP的$ N $子空间系统提供了各种新的必要条件。使用主表征定理，我们研究具有IBAP的子空间系统的属性，为子空间系统提供具有IBAP的系统的足够条件，并提供具有IBAP的子空间系统的示例。 这些结果应用于概率理论问题。令$（ω，\ MATHCAL {F}，μ）$为概率空间，$ \ Mathcal {f} _1，...，\ Mathcal {f} _n $ be sub- $σ$ -SALGEBRAS的$ \ Mathcal {F} $。我们会说，如果对于任意的随机变量$ξ_1，...，...，ξ_n$，（1）$ξ_k$是$之一（2）$ e |ξ_k|^2 <\ infty $，$ k = 1,2，...，n $; （3）$eξ_1=eξ_2= ... =eξ_n$，存在一个随机变量$ξ$，这样$ e | e | e |^2 <\ infty $和$ e（ξ| \ mathcal {f} _k）=ξ_k$ for ALL $ k = 1,2，...，...，n $。我们将证明，只有当相应的边缘子空间系统具有IBAP时，且只有在相应的边缘子空间系统中才具有IMP的集合。我们考虑两个例子。在第一个示例中，$ω= \ mathbb {n} $，在第二个示例中$ω= [a，b）$。对于这些示例，我们建立了IMP，IBAP，边缘子空间之和的闭合度与度量$μ$尾巴的“快速减小”之间的关系。
Let $H$ be a Hilbert space and $H_1,...,H_n$ be closed subspaces of $H$. Denote by $P_k$ the orthogonal projection onto $H_k$, $k=1,2,...,n$. Following Patrick L. Combettes and Noli N. Reyes, we will say that the system of subspaces $H_1,...,H_n$ possesses the inverse best approximation property (IBAP) if for arbitrary elements $x_1\in H_1,...,x_n\in H_n$ there exists an element $x\in H$ such that $P_k x=x_k$ for all $k=1,2,...,n$. We provide various new necessary and sufficient conditions for a system of $n$ subspaces to possess the IBAP. Using the main characterization theorem, we study properties of the systems of subspaces which possess the IBAP, obtain a sufficient condition for a system of subspaces to possess the IBAP, and provide examples of systems of subspaces which possess the IBAP. These results are applied to a problem of probability theory. Let $(Ω,\mathcal{F},μ)$ be a probability space and $\mathcal{F}_1,...,\mathcal{F}_n$ be sub-$σ$-algebras of $\mathcal{F}$. We will say that the collection $\mathcal{F}_1,...,\mathcal{F}_n$ possesses the inverse marginal property (IMP) if for arbitrary random variables $ξ_1,...,ξ_n$ such that (1) $ξ_k$ is $\mathcal{F}_k$-measurable, $k=1,2,...,n$; (2) $E|ξ_k|^2<\infty$, $k=1,2,...,n$; (3) $Eξ_1=Eξ_2=...=Eξ_n$, there exists a random variable $ξ$ such that $E|ξ|^2<\infty$ and $E(ξ|\mathcal{F}_k)=ξ_k$ for all $k=1,2,...,n$. We will show that a collection of sub-$σ$-algebras possesses the IMP if and only if the system of corresponding marginal subspaces possesses the IBAP. We consider two examples; in the first example $Ω=\mathbb{N}$, in the second example $Ω=[a,b)$. For these examples we establish relations between the IMP, the IBAP, closedness of the sum of marginal subspaces and "fast decreasing" of tails of the measure $μ$.
