# Unrestricted iterations of relaxed projections in Hilbert space: Regularity, absolute convergence, and statistics of displacements

## TL;DR

This paper investigates the behavior of relaxed projection iterations in Hilbert spaces, establishing uniform bounds on displacement moments and absolute convergence under mild regularity assumptions, regardless of relaxation parameters.

## Contribution

It proves that all trajectories from unrestricted relaxed projections have uniformly bounded displacement moments and absolute convergence, extending prior norm convergence results.

## Key findings

- Displacement series converges absolutely in Hilbert space.
- Uniform bounds on displacement moments for all trajectories.
- Effective bounds on the distribution of displacement norms.

## Abstract

Given a finite collection $\mathbf{V}:=(V_1,\dots,V_N)$ of closed linear subspaces of a real Hilbert space $H$, let $P_i$ denote the orthogonal projection operator onto $V_i$ and $P_{i,\lambda}:= (1-\lambda)I + \lambda P_i$ denote its relaxation with parameter $\lambda \in [0,2]$, $i=1,\dots,N$. Under a mild regularity assumption on $\mathbf{V}$ known as `innate regularity' (which, for example, is always satisfied if each $V_i$ has finite dimension or codimension), we show that all trajectories $(x_n)_{0}^\infty$ resulting from the iteration $x_{n+1} := P_{i_n,\lambda_n}(x_n)$, where the $i_n$ and the $\lambda_n$ are unrestricted other than the assumption that $\{\lambda_n : n \in \mathbb{N}\} \subset [\eta,2{-}\eta]$ for some $\eta \in (0,1]$, possess uniformly bounded displacement moments of arbitrarily small orders. In particular, we show that $$ \sum_{n=0}^\infty \|x_{n+1} - x_n \|^\gamma \leq C \|x_0\|^\gamma ~\mbox{ for all }~ \gamma > 0,$$ where $C:=C(\mathbf{V},\eta,\gamma)<\infty$. This result strengthens prior results on norm convergence of these trajectories, known to hold under the same regularity assumption. For example, with $\gamma=1$, it follows that the displacements series $\sum (x_{n+1}-x_n)$ converges absolutely in $H$. Quantifying the constant $C(\mathbf{V},\eta,\gamma)$, we also derive an effective bound on the distribution function of the norms of the displacements (normalized by the norm of the initial condition) which yields a root-exponential type decay bound on their decreasing rearrangement, again uniformly for all trajectories.