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

## TL;DR

This paper investigates the optimal error bounds for the initial step in cyclic alternating projections in Hilbert spaces, providing explicit formulas and bounds for the associated functions that measure convergence rates.

## Contribution

It introduces a novel approach linking the problem to Hermitian matrix optimization, deriving explicit formulas for three subspaces, and establishing bounds for general cases.

## Key findings

- Explicit formula for f_3(c) derived.
- Bounds for f_n(c) established for all n ≥ 4.
- Connection made between error bounds and Hermitian matrix optimization.

## Abstract

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.