Extending Meshulam's result on the boundedness of orbits of relaxed projections onto affine subspaces from finite to infinite-dimensional Hilbert spaces

TL;DR

This paper extends Meshulam's 1996 boundedness result for sequences generated by projections onto affine subspaces from finite-dimensional Euclidean spaces to infinite-dimensional Hilbert spaces, under certain regularity conditions.

Contribution

It generalizes Meshulam's theorem to infinite-dimensional Hilbert spaces using an induction approach based on the number of subspaces, with a key regularity assumption.

Findings

Boundedness of projection sequences in Hilbert spaces established

The regularity condition on linear subspaces is necessary

Connections made to randomized block Kaczmarz methods

Abstract

In 1996, Meshulam proved that any sequence generated in Euclidean space by randomly projecting onto affine subspaces drawn from a finite collection stays bounded even if the intersection of the subspaces is empty. His proof, which works even for relaxed projections, relies on an ingenious induction on the dimension of the Euclidean space. In this paper, we extend Meshulam's result to the general Hilbert space setting by an induction proof of the number of affine subspaces in the given collection. We require that the corresponding parallel linear subspaces are innately regular -- this assumption always holds in Euclidean space. We also discuss the sharpness of our result and make a connection to randomized block Kaczmarz methods.