Completeness, special functions and uncertainty principles over q-linear grids

Abstract

We derive completeness criteria for sequences of functions of the form $% f(x\lambda_{n})$, where $\lambda_{n}$ 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 $\{q^{-n}\}_{n=1}^{% \infty}$, $0<q<1$, then $f$ must vanish on the whole $q$-linear grid $% \{q^{n}\} _{n=-\infty}^{\infty}$.