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arXiv · 1707.05216

On Basic Fourier-Bessel Expansions

José Luis Cardoso
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Abstract

When dealing with Fourier expansions using the third Jackson (also known as Hahn-Exton) $q$-Bessel function, the corresponding positive zeros $j_{k\nu}$ and the "shifted" zeros, $qj_{k\nu}$, among others, play an essential role. Mixing classical analysis with $q$-analysis we were able to prove asymptotic relations between those zeros and the "shifted" ones, as well as the asymptotic behavior of the third Jackson $q$-Bessel function when computed on the "shifted" zeros. A version of a $q$-analogue of the Riemann-Lebesgue theorem within the scope of basic Fourier-Bessel expansions is also exhibited.

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José Luis Cardoso. 2017-07-10. On Basic Fourier-Bessel Expansions. https://doi.org/10.3842/sigma.2018.035

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