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On Basic Fourier-Bessel Expansions

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José Luis Cardoso

Classical Analysis and ODEs · Submitted 10 Jul 2017

abstract compiled · 2026-09-23 · no language-model review

The authors' main claim 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.

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arXiv ID1707.05216 (version 3)
Pith IDDMNMTAWA, Pith's permanent record of this paper and its review
CitationJos\'e Luis Cardoso. "On Basic Fourier-Bessel Expansions." arXiv:1707.05216 (2017). https://pith.science/paper/DMNMTAWA
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Compiled by classifier-1.13.0 via TypeSafe API, question bank abstract-p1-20260923, 2026-09-23: 1 request, 3,131 input tokens.

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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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#sentencerolethe authors' own workstates the main conclusionstrength
s0 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. background 1.00 0.09 0.06 asserted 1.00
s1 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. result 0.62 0.95 0.61 asserted 1.00
s2 A version of a $q$-analogue of the Riemann-Lebesgue theorem within the scope of basic Fourier-Bessel expansions is also exhibited. result 0.63 0.88 0.17 asserted 1.00

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Original 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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1 arXiv paper cite this in 4 sentences

Each citing sentence is taken from the citing paper's LaTeX source and typed by Pith's classifier. A label appears at probability 0.8 or more (0.95 for support), a contest at 0.7 or more; the quoted words are the citing authors'. How each label was checked · JSON

What it cites · 4 sentences without a confident label

Cites as background 4 papers
  • On this subject see [ref] and [ref] .

    The Nevanlinna parametrization for $q$-Lommel polynomials in the indeterminate case · Definitions and preliminary results · 0.97

  • Regarding criteria for completeness in $L_q[0,1]$ see [ref] , with an interesting application to a $q$-version of the uncertainty principle via the $q$-Hankel transform and a completeness property of the third Jackson $q$-Bessel function.

    math.CA/0602440 · Introduction · 0.91

  • Basic Fourier expansions were studied in [ref] , with respect to quadratic grids and to linear grids, respectively.

    math.CA/9706216 · Introduction · 0.99

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