Equivalent (Quasi)Norms for Certain Function Spaces of Generalized Mixed Smoothness

For certain (quasi)Banach function spaces of generalized mixed positive smoothness defined by (mixed) weighted norms of smooth dyadic decompositions of their Fourier transforms, characterizations are obtained in terms of rather general averages, their Peetre maximal functions, atomic and molecular representations, as well as in terms of the so-called $\varphi$-transforms.