Homeomorphisms and Fourier expansion

TL;DR

This paper surveys a recent result showing that for any continuous function, a suitable homeomorphism can be found to make its Fourier series converge uniformly, addressing a problem posed by Luzin.

Contribution

The paper presents a survey of a new result demonstrating the existence of homeomorphisms that ensure uniform convergence of Fourier series for continuous functions.

Findings

Existence of homeomorphisms transforming functions for uniform Fourier convergence

Historical context linking to Luzin's problem

Details of the proof of the main result

Abstract

We survey our recent result that for every continuous function there is an absolutely continuous homeomorphism such that the composition has a uniformly converging Fourier expansion. We mention the history of the problem, orginally stated by Luzin, and some details of the proof.