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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 chosenentire function. Using these criteria, we construct systems of nonorthogonalFourier-Bessel functions and their $q$-analogues, as well as other completesets of $q$-special functions. The completeness of certain sets of $q$-Besselfunctions is then used to prove that, if a function $f$ and its $q$-Hankeltransform both vanish at the points $\{q^{-n}\}_{n=1}^{% \infty}$, $0<q<1$,then $f$ must vanish on the whole $q$-linear grid $% \{q^{n}\}_{n=-\infty}^{\infty}$.Comment: 15 pages, final version (first and second introductory paragraphs  switched, an easier proof of the last theorem

Completeness, special functions and uncertainty principles overq-linear grids