Error Bounds for the Method of Simultaneous Projections with Infinitely Many Subspaces

TL;DR

This paper analyzes the convergence properties of the simultaneous projection method when applied to infinitely many subspaces in a Hilbert space, providing optimal error bounds and convergence criteria.

Contribution

It establishes the optimal linear convergence error bound in terms of Friedrichs angle and relates it to super-polynomial and polynomial convergence behaviors.

Findings

Optimal error bound for linear convergence expressed via Friedrichs angle

Relation of convergence rate to the dichotomy theorem and super-polynomial convergence

Conditions for polynomial convergence with specific starting points

Abstract

We investigate the properties of the simultaneous projection method as applied to countably infinitely many closed and linear subspaces of a real Hilbert space. We establish the optimal error bound for linear convergence of this method, which we express in terms of the cosine of the Friedrichs angle computed in an infinite product space. In addition, we provide estimates and alternative expressions for the above-mentioned number. Furthermore, we relate this number to the dichotomy theorem and to super-polynomially fast convergence. We also discuss polynomial convergence of the simultaneous projection method which takes place for particularly chosen starting points.