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∈AQ/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.