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On the inverse best approximation property of systems of subspaces of a Hilbert space | Ivan Feshchenko
; | Date: |
5 Feb 2020 | Abstract: | 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 $x_1in H_1,...,x_nin H_n$ 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 |
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