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

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.