Characterizations of the elements of generalized Hölder–Zygmund spaces by means of their representation

In this paper, we give three characterizations of the elements of generalized Hölder–Zygmund spaces. The first one, based on the Littlewood–Paley decomposition is already known, but the proof given here is much simpler. The second one, based on the wavelet decompositions generalizes a result obtained by Jaffard and Meyer. The third one uses generalized interpolation spaces. These results naturally extend the ones holding for the classical Hölder–Zygmund spaces.