Convergence of remote projections onto convex sets

TL;DR

This paper investigates the convergence of sequences generated by remote projections onto convex sets in a Hilbert space, establishing necessary and sufficient conditions for norm and weak convergence based on the properties of the sets and the parameters controlling the projections.

Contribution

It introduces a new convergence condition (T) for remote projections, generalizing previous results and clarifying the role of symmetry and the parameters in convergence behavior.

Findings

Condition (T) is necessary and sufficient for norm convergence.

Sum of squared parameters diverging ensures weak convergence.

Examples show symmetry can be relaxed under certain conditions.

Abstract

Let $\{C_{\alpha}\}_{\alpha\in \Omega}$ be a family of closed and convex sets in a Hilbert space $H$, having a nonempty intersection $C$. We consider a sequence $\{x_n\}$ of remote projections onto them. This means, $x_0\in H$, and $x_{n+1}$ is the projection of $x_n$ onto such a set $C_{\alpha(n)}$ that the ratio of the distances from $x_n$ to this set and to any other set from the family is at least $t_n\in [0,1]$. We study properties of the weakness parameters $t_n$ and of the sets $C_\alpha$ which ensure the norm or weak convergence of the sequence $\{x_n\}$ to a point in $C$. We show that condition (T) is necessary and sufficient for the norm convergence of $x_n$ to a point in $C$ for any starting element and any family of closed, convex, and symmetric sets $C_\alpha$. This generalizes a result of Temlyakov who introduced (T) in the context of greedy approximation theory. We give examples explaining to what extent the symmetry condition on the sets $C_{\alpha}$ can be dropped. Condition (T) is stronger than $\sum t_n^2=\infty$ and weaker than $\sum t_n/n=\infty$. The condition $\sum t_n^2=\infty$ turns out to be necessary and sufficient for the sequence $\{x_n\}$ to have a partial weak limit in $C$ for any family of closed and convex sets $C_\alpha$ and any starting element.