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| Main Authors: | , |
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| Format: | Preprint |
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2020
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| Subjects: | |
| Online Access: | https://arxiv.org/abs/2005.06076 |
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| _version_ | 1866915964688269312 |
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| author | Uriostegui, Kenan Wolf, Kurt Bernardo |
| author_facet | Uriostegui, Kenan Wolf, Kurt Bernardo |
| contents | We present a straightforward discretization of the Bessel functions $J_n(x)$ to discrete counterparts $B^{(N)}_n(x_m)$, of $N$ integer orders $n$ on $N$ integer points $x_m \equiv m$, that we call discrete Bessel functions. These are built from a Bessel integral generating function, restricting the Fourier transform over the circle to $N$ points. We show that the discrete Bessel functions satisfy several linear and quadratic relations, particularly Graf's product-displacement formulas, that are exact analogues of well-known relations between the continuous functions. It is noteworthy that these discrete Bessel functions approximate very closely the values of the continuous functions in ranges $n + |m| < N$. For fixed $N$, this provides an $N$-point transform between functions of order and of position,$f_n$ and $\widetilde{f}_m$, which is efficient for the Fourier analysis of finite decaying signals. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2005_06076 |
| institution | arXiv |
| publishDate | 2020 |
| record_format | arxiv |
| spellingShingle | Discrete Bessel functions and transform Uriostegui, Kenan Wolf, Kurt Bernardo Mathematical Physics We present a straightforward discretization of the Bessel functions $J_n(x)$ to discrete counterparts $B^{(N)}_n(x_m)$, of $N$ integer orders $n$ on $N$ integer points $x_m \equiv m$, that we call discrete Bessel functions. These are built from a Bessel integral generating function, restricting the Fourier transform over the circle to $N$ points. We show that the discrete Bessel functions satisfy several linear and quadratic relations, particularly Graf's product-displacement formulas, that are exact analogues of well-known relations between the continuous functions. It is noteworthy that these discrete Bessel functions approximate very closely the values of the continuous functions in ranges $n + |m| < N$. For fixed $N$, this provides an $N$-point transform between functions of order and of position,$f_n$ and $\widetilde{f}_m$, which is efficient for the Fourier analysis of finite decaying signals. |
| title | Discrete Bessel functions and transform |
| topic | Mathematical Physics |
| url | https://arxiv.org/abs/2005.06076 |