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 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–Pál theorem. In the present work we propose a short and technically very simple proof of the Jurkat–Waterman theorem. Our approach yields a stronger result.