Completeness, special functions and uncertainty principles overq-linear grids
Luís Daniel Abreu · Journal of Physics A Mathematical and General · 2006
We derive completeness criteria for sequences of functions of the form f ( x λ n ), where λ n is the n th zero of a suitably chosen entire function. Using these criteria, we construct complete nonorthogonal systems of Fourier–Bessel functions and their q -analogues, as well as other complete sets of q -special functions. We discuss connections with uncertainty principles over q -linear grids and the completeness of certain sets of q -Bessel functions is 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 } ∞ n =−∞ .