Harmonic analysis on rational numbers

Harmonic analysis on Q is studied. The Fourier transforms of complex functions f ( y ) ( y ∈ Q ) in the Schwartz–Bruhat space S , are functions F ( u ) where u ∈ A Q / Q . It is shown that the corresponding Heisenberg–Weyl group is a locally compact topological group. Various time–frequency analysis methods (properties of displacement operators, Wigner and Ambiguity functions, etc.) are discussed in this context.