A short and simple proof of the Jurkat--Waterman theorem on conjugate functions

TL;DR

This paper presents a concise and simple proof of the Jurkat--Waterman theorem on conjugate functions, enhancing the classical Bohr--Pál theorem by leveraging a new approach.

Contribution

It offers a shorter, more straightforward proof of the Jurkat--Waterman theorem and extends its results with a stronger conclusion.

Findings

Simplified proof of the Jurkat--Waterman theorem

Improved results over the classical Bohr--Pál theorem

Methodology applicable to related Fourier analysis problems

Abstract

It is well--known that certain properties of continuous functions on the circle $\mathbb T$, related to the Fourier expansion, can be improved by a change of variable, i.e., by a homeomorphism of the circle onto itself. One of the results in this area is the Jurkat--Waterman theorem on conjugate functions, which improves the classical Bohr--P\'al theorem. In the present work we provide a short and technically very simple proof of the Jurkat--Waterman theorem. Our approach yields a stronger result.