Ergodic behaviour of a Douglas-Rachford operator away from the origin

TL;DR

This paper demonstrates that the Douglas-Rachford operator, when away from the origin, can be approximated by an operator satisfying a weak ergodic theorem, extending to other projection and reflection operators.

Contribution

It introduces a novel approximation of the Douglas-Rachford operator away from the origin that satisfies a weak ergodic theorem, with implications for related operators.

Findings

Approximation of Douglas-Rachford operator satisfies weak ergodic theorem

Results extend to other projection and reflection operators

Provides new insights into ergodic behavior away from the origin

Abstract

It is shown that away from the origin, the Douglas-Rachford operator with respect to a sphere and a convex set in a Hilbert space can be approximated by a another operator which satisfies a weak ergodic theorem. Similar results for other projection and reflection operators are also discussed.