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

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

2006-02-20

arxiv.org/abs/math/0602440v2

J. Phys. A: Math. Gen. 39 (2006) 14567-14580

Mathematics

Classical Analysis and ODEs

15 pages, final version (first and second introductory paragraphs switched, an easier proof of the last theorem)

Scientific paper

10.1088/0305-4470/39/47/004

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

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