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

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.
The manuscript has been accepted for publication in Journal of Approximation Theory.