  In this work we construct harmonic analysis on free Abelian groups of rank
$2$, namely: we construct and investigate spaces of functions and
distributions, Fourier transforms, actions of discrete and extended discrete
Heisenberg groups. In case of the rank-$2$ value group of a two-dimensional
local field with finite last residue field we connect this harmonic analysis
with harmonic analysis on the two-dimensional local field, where the latter
harmonic analysis was constructed in earlier works by the authors.
