Science-advisor
REGISTER info/FAQ
Login
username
password
     
forgot password?
register here
 
Research articles
  search articles
  reviews guidelines
  reviews
  articles index
My Pages
my alerts
  my messages
  my reviews
  my favorites
 
 
Stat
Members: 3643
Articles: 2'488'730
Articles rated: 2609

29 March 2024
 
  » arxiv » 2002.1959

 Article overview


On the inverse best approximation property of systems of subspaces of a Hilbert space
Ivan Feshchenko ;
Date 5 Feb 2020
AbstractLet $H$ 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 $x_1in H_1,...,x_nin H_n$ there exists an element $xin H$ 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 $(Omega,mathcal{F},mu)$ be a probability space and $mathcal{F}_1,...,mathcal{F}_n$ be sub-$sigma$-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 $xi_1,...,xi_n$ such that (1) $xi_k$ is $mathcal{F}_k$-measurable, $k=1,2,...,n$; (2) $E|xi_k|^2<infty$, $k=1,2,...,n$; (3) $Exi_1=Exi_2=...=Exi_n$, there exists a random variable $xi$ such that $E|xi|^2<infty$ and $E(xi|mathcal{F}_k)=xi_k$ for all $k=1,2,...,n$. We will show that a collection of sub-$sigma$-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 $Omega=mathbb{N}$, in the second example $Omega=[a,b)$. 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 $mu$.
Source arXiv, 2002.1959
Services Forum | Review | PDF | Favorites   
 
Visitor rating: did you like this article? no 1   2   3   4   5   yes

No review found.
 Did you like this article?

This article or document is ...
important:
of broad interest:
readable:
new:
correct:
Global appreciation:

  Note: answers to reviews or questions about the article must be posted in the forum section.
Authors are not allowed to review their own article. They can use the forum section.

browser claudebot






ScienXe.org
» my Online CV
» Free


News, job offers and information for researchers and scientists:
home  |  contact  |  terms of use  |  sitemap
Copyright © 2005-2024 - Scimetrica