Constructive approach to the monotone rearrangement of functions

TL;DR

This paper presents a simple, convergent method for constructing the monotone rearrangement of functions from samples, with practical algorithms for continuous functions and theoretical insights for measurable functions.

Contribution

It introduces a straightforward procedure for monotone rearrangement that converges in the continuous case and discusses its theoretical extension to measurable functions.

Findings

Converges to the monotone rearrangement for continuous functions.

Uniform convergence under additional assumptions.

Applicable to measurable functions with theoretical interest.

Abstract

We detail a simple procedure (easily convertible to an algorithm) for constructing from quasi-uniform samples of $f$ a sequence of linear spline functions converging to the monotone rearrangement of $f$, in the case where $f$ is an almost everywhere continuous function defined on a bounded set $\Omega$ with negligible boundary. Under additional assumptions on $f$ and $\Omega$, we prove that the convergence of the sequence is uniform. We also show that the same procedure applies to arbitrary measurable functions too, but with the substantial difference that in this case the procedure has only a theoretical interest and cannot be converted to an algorithm.