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The Finite Bessel Transforms |
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PP: 723-730 |
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doi:10.18576/amis/150606
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Author(s) |
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Kenan Uriostegui,
Kurt Bernardo Wolf,
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Abstract |
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| Special functions that are generated by a Fourier transform
over a circle, also provide {\it discrete\/} counterparts,
where the circle is substituted by $N$ equidistant points
over that circle, with the finite Fourier transform over them.
This process was applied to Bessel and Mathieu functions in
[{\it Appl.\ Math.\ Inf.\ Sci.} {\bf 15}, {307--315} (2021)].
The resulting {\it discrete Bessel functions},
$B_n^\ssty{N}(x)$, $n\in\{0,1,\ldots,N{-}1\}$, satisfy the
linear and Graf quadratic relations of their continuous
counterparts, and provide a very close numerical approximation
with $\lim_{N\to\infty}B_n^\ssty{N}(x) = J_n(x)$.
In this paper, the $N\times N$ matrices ${\bf B}=\Vert B_{n,m}\Vert$,
for $B_{n,m}:=B_n^\ssty{N}(x_m)$ over $x_m\in\{0,1,\ldots,\,N{-}1\}$,
are used to define transform kernels between $N$ functions of
position $f_m$ and of Bessel mode $\widetilde f_n$,
which are efficient for the Fourier analysis of discrete signals
with $f_m\propto m^{-1/2}$ decay. |
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