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

Luis Daniel Abreu
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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 $% \{q^{n}\} _{n=-\infty}^{\infty}$.

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Luis Daniel Abreu. 2006-02-20. Completeness, special functions and uncertainty principles over q-linear grids. https://doi.org/10.1088/0305-4470/39/47/004

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