Bessel transform

Bessel Transform or BesselTransform, called also Hankel transform

at order $\nu$ is operator that converts function $f$ to funciton $g=\mathrm{BesselTransform}_\nu(f)$ such that

$\!\!\!\!\!\!\!\!\!\!\! (1) \displaystyle ~ ~ ~

g ( p ) =\int_0^\infty \mathrm{BesselJ}_\nu(px) f(x) x ~\mathrm d x$ where BesselJ$_\nu$ is the Bessel function, and $\nu$ is assumed to be real number; in many applications $\nu$ is natural number or just zero.

No indications of any difference between meanings of terms Hankel transform and Bessel transform is found.

The second iteration of the BesselTransform is Identity transform, id set, $\mathrm{BesselTransform}^2= $IdentiryOperator, and, for $f$ and $g$ related with (1), the inverse relation takes place,

$\!\!\!\!\!\!\!\!\!\!\! (2) \displaystyle ~ ~ ~

f ( p ) =\int_0^\infty \mathrm{BesselJ}_\nu(px) g(x) x ~ \mathrm d x$

at least for continuous functions $f$ and $g$.

WARNING! The formulas of this article are not yet verified with pics, so, test well all you need before to use it! In addition, the kernel is not the same as that from GNU, copypasted in Discrete Hankel transform. It should be revealed, which kernel is "correct".

Discrete Bessel

Te discrete analogy of the Bessel transform is defined at an array $f$ of length $N$, and gives array $g=\mathrm{BesselTransform}_{N,\nu}(f)$ with the following formula:

$\!\!\!\!\!\!\!\!\!\!\! (3) \displaystyle ~ ~ ~

g_m = \sum_{n=1}^{N}~ T_{m,n}~ f_n$ with matrix $T$ defined with

$\!\!\!\!\!\!\!\!\!\!\! (4) \displaystyle ~ ~ ~

T_{m,n}=\frac{ 2~ \mathrm{BesselJ}_{\nu} \left( \frac{ \alpha_{\nu,n}~ \alpha_{\nu,m} }{\alpha_{\nu,N+1}} \right) }{ \left| \mathrm{BesselJ}_{\nu+1}(\alpha_{\nu,m}) \right|~ \left| \mathrm{BesselJ}_{\nu+1}(\alpha_{\nu,n}) \right|~ \alpha_{\nu,N+1} }$ where $\alpha_{\nu,n}$ is $n$th zero of function BesselJ$_{\nu}$.

The $n$th element of array $f$ or $g$ corresponds to the function evaluated at point $\displaystyle \frac{\alpha_{\nu, n}}{\sqrt{\alpha_{\nu,N}}}$

Scaling of the argument

As in the case of the Fourier transform, the scale of the arguments of the functions $f$ and $g$ can be varied, defining the new function $ F(x)=f(x ~ M) M$ and $ G(x)=g( x / M) /M$ where $M$ is real parameter of magnification. Similar scaling can be applied also to the discrete Bessel transform.

Scaling of the function

The additional factor in the integral representation of the Bessel transform can be eliminated, multiplying each, the function and its transform, to the square root of the argument, with corresponding modification of the kernel. .

Truncated Hankel transform

The "truncated Hankel transform" called also "finite Hanlel transform" relates functions $f$ and $g$ such that

$ \displaystyle

f(x)= \int_0^1 J_m(xy)~ g(y)~y ~\mathrm{d} y $ In notations by , the functions are already multiplied by the square toot of their argument; for example, $T(x)=f(x)\sqrt{x}$.

The corresponding modification of the kernel of the integral transfom, $\mathrm d y$ appears instead of $y\,\mathrm d y$. It is not clear which notation is better; so, perhaps, both notations will be used in TORI.

However, the second iteration of the truncated Hankel transform is not the identity transform.

Application of the Bessel transform

The Bessel transform of zero-th order (id set, $\nu=0$ in the formulas above) is useful for the description of propagation of waves with circular (radial) symmetty.

References

http://dsp-book.narod.ru/HFTSP/8579ch17.pdf Poularikas A. D. “The Hankel Transform” The Handbook of Formulas and Tables for Signal Processing. Ed. Alexander D. Poularikas. Boca Raton: CRC Press LLC,1999.

http://en.wikipedia.org/wiki/Hankel_transform

http://www.acms.arizona.edu/FemtoTheory/MK_personal/opti547/literature/QD-Hankel.pdf M.Guizar-Sicairos, J.C.Gutierrez-Vega. Computation of quasi-discrete Hankel transforms of integer order for propagating optical wave fields. Vol. 21, No. 1/January 2004/J.Opt.Soc.Am.A, p.53-58.

http://www.network-theory.co.uk/docs/gslref/DiscreteHankelTransformDefinition.html GNU Scientific Library Reference Manual - Third Edition (v1.12) by M. Galassi, J. Davies, J. Theiler, B. Gough, G. Jungman, P. Alken, M. Booth, F. Rossi

$ \displaystyle g_m = \int_0^1 t \mathrm d t ~ J_\nu(j_{\nu,m}t) f(t)$

$ \displaystyle f(t) = \sum_{m=1}^\infty (2 J_\nu(j_{\nu,m}x) / J_{\nu+1}(j_{\nu,m})^2) g_m ~ ~ ~$ (perhaps, $t=x$ )

$ \displaystyle

g_m = (2 / j_{\nu,M}^2)

     \sum_{k=1}^{M-1} f(j_{\nu,k}/j_{\nu,M})
         (J_\nu(j_{\nu,m} j_{\nu,k} / j_{\nu,M}) / J_{\nu+1}(j_{\nu,k})^2)

$

http://www.fh.huji.ac.il/~ronnie/Papers/k31.pdf JOURNAL OF COMPUTATIONAL PHYSICS 59, 136-151 (1985) ROB BISSELINC AND RONNIE KOSLOFF. The Fast Hankel Transform as a Tool in the Solution of the Time Dependent Schrijdinger Equation.

Keywords

BesselJ0, Fourier transform