Equivalent Normings of Spaces of Functions of Variable Smoothness

For the Banach spaces $B_{p,q}^a$ and $F_{p,q}^a$ of functions defined on $\mathbb R^n$ whose variable smoothness $a=a(x)$ is determined by the behavior of their differences, equivalent normings are established in terms of weighted norms of smooth dyadic decompositions of their Fourier transforms.