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In mathematics, the Hankel transform expresses any given function f(r) as the weighted sum of an infinite number of Bessel functions of the first kind Jν(kr). The Bessel functions in the sum are all of the same order ν, but differ in a scaling factor k along the r-axis. The necessary coefficient Fν of each Bessel function in the sum, as a function of the scaling factor k constitutes the transformed function.
More precisely, the Hankel transform of order ν of a function f(r) is given by:
where Jν is the Bessel function of the first kind of order ν with ν ≥ −1/2. The inverse Hankel transform of Fν(k) is defined as:
which can be readily verified using the orthogonality relationship described below. The Hankel transform is an integral transform and was first developed by the mathematician Hermann Hankel. It is also known as the Fourier–Bessel transform. Just as the Fourier transform for an infinite interval is related to the Fourier series over a finite interval, so the Hankel transform over an infinite interval is related to the Fourier–Bessel series over a finite interval.
The Hankel transform of a function f(r) is valid at every point at which f(r) is continuous provided that the function is defined in (0, ∞), is piecewise continuous and of bounded variation in every finite subinterval in (0, ∞), and the integral
is finite. However, like the Fourier Transform, the domain can be extended by a density argument to include some functions whose above integral is not finite, for example ; this extension will not be discussed in this article.
The Bessel functions form an orthogonal basis with respect to the weighting factor r:
for k and k' greater than zero.
If f(r) and g(r) are such that their Hankel transforms Fν(k) and Gν(k) are well defined, then the Plancherel theorem states
Parseval's theorem, which states:
is a special case of the Plancherel theorem. These theorems can be proven using the orthogonality property.
The Hankel transform of order zero is essentially the two dimensional Fourier transform of a circularly symmetric function.
Consider a two-dimensional function f(r) of the radius vector r. Its Fourier transform is:
With no loss of generality, we can pick a polar coordinate system (r, θ) such that the k vector lies on the θ = 0 axis. The Fourier transform is now written in these polar coordinates as:
where θ is the angle between the k and r vectors. If the function f happens to be circularly symmetric, it will have no dependence on the angular variable θ and may be written f(r). The integration over θ may be carried out, and the Fourier transform is now written:
which is just 2π times the zero-order Hankel transform of f(r). For the reverse transform,
so f(r) is 1/2π times the zero-order Hankel transform of F(k).
To generalize: If f is expanded in a multipole series,
and if is the angle between the direction of k and the axis,
one may substitute ,
If is sufficiently smooth near the origin and zero outside a radius , it may be expanded into a Chebyshev series,
The numerically important aspect is that the expansion coefficients are accessible with Discrete Fourier transform techniques. Insertion into the previous formula yields
This is one flavor of fast Hankel transform techniques.
The Hankel transform is one member of the FHA cycle of integral operators. In two dimensions, if we define A as the Abel transform operator, F as the Fourier transform operator and H as the zeroth order Hankel transform operator, then the special case of the projection-slice theorem for circularly symmetric functions states that:
In other words, applying the Abel transform to a 1-dimensional function and then applying the Fourier transform to that result is the same as applying the Hankel transform to that function. This concept can be extended to higher dimensions.
|, z may be any complex number|
The Hankel transform of Zernike polynomials are essentially Bessel Functions (Noll 1976):
for even .
- Gaskill, Jack D. (1978). Linear Systems, Fourier Transforms, and Optics. New York: John Wiley & Sons. ISBN 0-471-29288-5.
- Polyanin, A. D.; Manzhirov, A. V. (1998). Handbook of Integral Equations. Boca Raton: CRC Press. ISBN 0-8493-2876-4.
- Smythe, William R. (1968). Static and Dynamic Electricity (3rd ed.). New York: McGraw-Hill. pp. 179–223.
- Offord, A. C. (1935). "On Hankel transforms". Proceedings of the London Mathematical Society 39 (2): 49–67. doi:10.1112/plms/s2-39.1.49.
- Eason, G.; Noble, B.; Sneddon, I. N. (1955). "On certain integrals of Lipschitz-Hankel type involving products of Bessel Functions". Philosophical Transactions of the Royal Society A 247 (935): 529–551. JSTOR 91565.
- Kilpatrick, J. E.; Katsura, Shigetoshi; Inoue, Yuji (1967). "Calculation of integrals of products of Bessel functions". Mathematics of Computation 21 (99): 407–412. doi:10.1090/S0025-5718-67-99149-1.
- MacKinnon, Robert F. (1972). "The asymptotic expansions of Hankel transforms and related integrals". Mathematics of Computation 26 (118): 515–527. doi:10.1090/S0025-5718-1972-0308695-9. JSTOR 2003243.
- Linz, Peter; Kropp, T. E. (1973). "A note on the computation of integrals involving products of trigonometric and Bessel functions". Mathematics of Computation 27 (124): 871–872. JSTOR 2005522.
- Noll, Robert J (1976). "Zernike polynomials and atmospheric turbulence". Journal of the Optical Society of America 66 (3): 207–211. Bibcode:1976JOSA...66..207N. doi:10.1364/JOSA.66.000207.
- Siegman, A. E. (1977). "Quasi-fast Hankel transform". Opt. Lett. 1 (1): 13–15. Bibcode:1977OptL....1...13S. doi:10.1364/OL.1.000013.
- Magni, Vittorio; Cerullo, Giulio; De Silverstri, Sandro (1992). "High-accuracy fast Hankel transform for optical beam propagation". J. Opt. Soc. Am. A 9 (11): 2031–2033. Bibcode:1992JOSAA...9.2031M. doi:10.1364/JOSAA.9.002031.
- Agnesi, A.; Reali, Giancarlo C.; Patrini, G.; Tomaselli, A. (1993). "Numerical evaluation of the Hankel transform: remarks". Journal of the Optical Society of America A 10 (9): 1872. Bibcode:1993JOSAA..10.1872A. doi:10.1364/JOSAA.10.001872.
- Barakat, Richard (1996). "Numerical evaluation of the zero-order Hankel transform using Filon quadrature philosophy". Applied Mathematics Letters 9 (5): 21–26. MR 1415467.
- Ferrari, José A.; Perciante, Daniel; Dubra, Alfredo (1999). "Fast Hankel transform of nth order". J. Opt. Soc. Am. A 16 (10): 2581–2582. Bibcode:1999JOSAA..16.2581F. doi:10.1364/JOSAA.16.002581.
- Secada, José D. (1999). "Numerical evaluation of the Hankel transform". Comp. Phys. Comm. 116 (2–3): 278–294. Bibcode:1999CoPhC.116..278S. doi:10.1016/S0010-4655(98)00108-8.
- Wieder, Thomas (1999). "Algorithm 794: Numerical Hankel transform by the Fortran program HANKEL". ACM Trans. Math. Soft. 25 (2): 240–250. doi:10.1145/317275.317284.
- Knockaert, Luc (2000). "Fast Hankel transform by fast sine and cosine transforms: the Mellin connection". IEEE Trans. Signal Process. 48 (6): 1695–1701.
- Zhang, D. W.; Yuan, X.-C.; Ngo, N. Q.; Shum, P. (2002). "Fast Hankel transform and its application for studying the propagation of cylindrical electromagnetic fields". Opt. Exp. 10 (12): 521–525.
- Markham, Joanne; Conchello, Jose-Angel (2003). "Numerical evaluation of Hankel transforms for oscillating functions". J. Opt. Soc. Am. A 20 (4): 621–630. Bibcode:2003JOSAA..20..621M. doi:10.1364/JOSAA.20.000621.
- Perciante, César D.; Ferrari, José A. (2004). "Fast Hankel transform of nth order with improved performance". J. Opt. Soc. Am. A 21 (9): 1811. Bibcode:2004JOSAA..21.1811P. doi:10.1364/JOSAA.21.001811.
- Gizar-Sicairos, Manuel; Guitierrez-Vega, Julio C. (2004). "Computation of quasi-discrete Hankel transform of integer order for propagating optical wave fields". J. Opt. Soc. Am. A 21 (1): 53–58. Bibcode:2004JOSAA..21...53G. doi:10.1364/JOSAA.21.000053.
- Cerjan, Charles (2007). "The Zernike-Bessel representation and its application to Hankel transforms". J. Opt. Soc. Am. A 24 (6): 1609–1616. Bibcode:2007JOSAA..24.1609C. doi:10.1364/JOSAA.24.001609.