We define and characterize weak and strong two-scale convergence in Lp
0 and other spaces via a transformation of variable, extending Nguetseng's definition.
We derive several properties, including weak and strong two-scale compactness;
in particular we prove two-scale versions of theorems of
Ascoli-Arzelà, Chacon, Riesz, and Vitali.
We then approximate two-scale derivatives, and define two-scale convergence in
spaces of either weakly or strongly differentiable functions.
We also derive two-scale versions of the classic theorems of Rellich, Sobolev, and Morrey.