On the optimal error bound for the first step in the method of cyclic alternating projections

Let \(H\) be a Hilbert space and \(H_1,...,H_n\) be closed subspaces of \(H\). Set \(H_0:=H_1\cap H_2\cap...\cap H_n\) and let \(P_k\) be the orthogonal projection onto \(H_k\), \(k=0,1,...,n\). The paper is devoted to the study of functions \(f_n:[0,1]\to\mathbb{R}\) defined by $$ f_n(c)=\sup\{\|P_...

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Bibliographic Details
Published in:arXiv.org 2019-08
Main Author: Feshchenko, Ivan
Format: Article
Language:English
Subjects:
Equivalence
Hilbert space
Optimization
Subspaces
Upper bounds
Online Access:Get full text
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Summary:Let \(H\) be a Hilbert space and \(H_1,...,H_n\) be closed subspaces of \(H\). Set \(H_0:=H_1\cap H_2\cap...\cap H_n\) and let \(P_k\) be the orthogonal projection onto \(H_k\), \(k=0,1,...,n\). The paper is devoted to the study of functions \(f_n:[0,1]\to\mathbb{R}\) defined by $$ f_n(c)=\sup\{\|P_n...P_2 P_1-P_0\|\,|c_F(H_1,...,H_n)\leqslant c\},\,c\in[0,1], $$ where the supremum is taken over all systems of subspaces \(H_1,...,H_n\) for which the Friedrichs number \(c_F(H_1,...,H_n)\) is less than or equal to \(c\). Using the functions \(f_n\) one can easily get an upper bound for the rate of convergence in the method of cyclic alternating projections. We will show that the problem of finding \(f_n(c)\) is equivalent to a certain optimization problem on a subset of the set of Hermitian complex \(n\times n\) matrices. Using the equivalence we find \(f_3\) and study properties of \(f_n\), \(n\geqslant 4\). Moreover, we show that $$ 1-a_n(1-c)-\widetilde{b}_n(1-c)^2\leqslant f_n(c)\leqslant 1-a_n(1-c)+b_n(1-c)^2 $$ for all \(c\in[0,1]\), where \(a_n=2(n-1)\sin^2(\pi/(2n))\), \(b_n=6(n-1)^2\sin^4(\pi/(2n))\) and \(\widetilde{b}_n\) is some positive number.
ISSN:2331-8422