Verhalten der Cauchy-transformation und der Hilbert-transformation für auf dem einheitskreis stetige funktionen

In this paper we investigate the behavior of the Hilbert transform and the Cauchy transform. It is well known, that for absolut integrable functions the Hilbert transform and the Cauchy transform is finite almost everywhere. In this paper it is shown, that for each set E ⊂ [-π, π) with Lebesgue measure zero there exists a continuous function such that the Hilbert transform and the Cauchy transform of this function is infinite for all points of the set E. So for continuous functions the Hilbert transform and the Cauchy transform have a similar divergence behavior as for absolute integrable functions.