. C A ] 2 0 Fe b 20 06 Completeness , special functions and uncertainty principles over q - linear grids Lúıs Daniel Abreu

We derive completeness criteria for sequences of functions of the form f(xλn), where λn is the nth zero of a suitably chosen entire function. Using these criteria, we construct systems of nonorthogonal Fourier-Bessel functions and their q-analogues, as well as other complete sets of q-special functions. The completeness of certain sets of q-Bessel functions is then used to prove that, if a function f and its q-Hankel transform both vanish at the points {q−n}∞ n=1, 0 < q < 1, then f must vanish on the whole q-linear grid {q}∞ n=−∞.