Applied Mathematics & Information Sciences

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Me ́xico


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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