# Hankel transform

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## Hankel transform

[′häŋk·əl ‚tranz‚fȯrm]
(mathematics)
The Hankel transform of order m of a real function ƒ(t) is the function F (s) given by the integral from 0 to ∞ of ƒ(t) tJ m (st) dt, where J m denotes the m th-order Bessel function. Also known as Bessel transform; Fourier-Bessel transform.
McGraw-Hill Dictionary of Scientific & Technical Terms, 6E, Copyright © 2003 by The McGraw-Hill Companies, Inc.
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If d = 1 and G = [Z.sub.2], then the Dunkl transform is related closely to the Hankel transform on the real line.
It is noted that [v.sub.n](t) is zero-order finite Hankel transform of function v(r, t) and [u.sub.n](t) is the finite Hankel transform of the first order of function u(r, t), respectively.
The Hankel transform (the sequence of determinants of Hankel matrixes) ([d.sub.n]; n [member of] N) reads
Combining (29) and (32), we can get (33) based on Hankel transform:
Further theories and applications of this matrix had been established including the Hankel determinant and Hankel transform. The determinant of the Hankel matrix is called Hankel determinant, while the sequence of Hankel determinants is called Hankel transform as defined by Aigner [19].
Fan, "Fractional Hankel transform studied by charge-amplitude state representations and complex fractional Fourier transformation," Optics Letters, vol.
Pandey, "Numerical evaluation of the Hankel transform by using linear Legendre multi-wavelets," Computer Physics Communications, vol.
The Hankel transform of an integer sequence and some of its properties were discussed by Layman in [11].
The properties of the elegant Laguerre-Gaussian beams propagating in free space [2-4], through a paraxial ABCD optical system [5], in apertured fractional Hankel transform systems [6], through aligned and misaligned paraxial optical systems [7], at a dielectric interface [8], in turbulent atmosphere [9], in non-Kolmogorov turbulence [10], by an opaque obstacle [11], and in a uniaxial crystal [12] have been extensively investigated.
First employing Hankel transform with respect to the variable r, which is denoted by * and then Laplace transform with respect to the variable z, which is denoted by--.
In this paper, we study the properties of the eigenfunctions of the finite Hankel transform. We deduce a sampling series in terms of these functions for Hankel-band-limited signals and derive bounds for the truncation error of the sampling series.

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