The Finite Bessel Transforms

Authors

Kenan Uriostegui, Institute of Physical Sciences, National Autonomous University of Me ́xico, Morelos 62251, Me ́xicoFollow
Kurt Bernardo Wolf, Institute of Physical Sciences, National Autonomous University of Me ́xico, Morelos 62251, Me ́xicoFollow

Author Country (or Countries)

Me ́xico

Abstract

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.

Digital Object Identifier (DOI)

http://dx.doi.org/10.18576/amis/150606

Recommended Citation

Uriostegui, Kenan and Bernardo Wolf, Kurt (2021) "The Finite Bessel Transforms," Applied Mathematics & Information Sciences: Vol. 15: Iss. 6, Article 6.
DOI: http://dx.doi.org/10.18576/amis/150606
Available at: https://digitalcommons.aaru.edu.jo/amis/vol15/iss6/6