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33.1 Definitions

The discrete Hankel transform acts on a vector of sampled data, where the samples are assumed to have been taken at points related to the zeros of a Bessel function of fixed order; compare this to the case of the discrete Fourier transform, where samples are taken at points related to the zeroes of the sine or cosine function.

Starting with its definition, the Hankel transform (or Bessel transform) of order \nu of a function f with \nu > -1/2 is defined as (see Johnson, 1987 and Lemoine, 1994)

F_\nu(u) = \int_0^\infty f(t) J_\nu(u t) t dt

If the integral exists, F_\nu is called the Hankel transformation of f. The reverse transform is given by

f(t) = \int_0^\infty F_\nu(u) J_\nu(u t) u du ,

where \int_0^\infty f(t) t^{1/2} dt must exist and be absolutely convergent, and where f(t) satisfies Dirichlet’s conditions (of limited total fluctuations) in the interval [0,\infty].

Now the discrete Hankel transform works on a discrete function f, which is sampled on points n=1...M located at positions t_n=(j_{\nu,n}/j_{\nu,M}) X in real space and at u_n=j_{\nu,n}/X in reciprocal space. Here, j_{\nu,m} are the m-th zeros of the Bessel function J_\nu(x) arranged in ascending order. Moreover, the discrete functions are assumed to be band limited, so f(t_n)=0 and F(u_n)=0 for n>M. Accordingly, the function f is defined on the interval [0,X].

Following the work of Johnson, 1987 and Lemoine, 1994, the discrete Hankel transform is given by

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

It is this discrete expression which defines the discrete Hankel transform calculated by GSL. In GSL, forward and backward transforms are defined equally and calculate F_\nu(u_m). Following Johnson, the backward transform reads

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

Obviously, using the forward transform instead of the backward transform gives an additional factor X^4/j_{\nu,M}^2=t_m^2/u_m^2.

The kernel in the summation above defines the matrix of the \nu-Hankel transform of size M-1. The coefficients of this matrix, being dependent on \nu and M, must be precomputed and stored; the gsl_dht object encapsulates this data. The allocation function gsl_dht_alloc returns a gsl_dht object which must be properly initialized with gsl_dht_init before it can be used to perform transforms on data sample vectors, for fixed \nu and M, using the gsl_dht_apply function. The implementation allows to define the length X of the fundamental interval, for convenience, while discrete Hankel transforms are often defined on the unit interval instead of [0,X].

Notice that by assumption f(t) vanishes at the endpoints of the interval, consistent with the inversion formula and the sampling formula given above. Therefore, this transform corresponds to an orthogonal expansion in eigenfunctions of the Dirichlet problem for the Bessel differential equation.

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