Discrete Hankel Transforms

This chapter describes functions for performing Discrete Hankel Transforms (DHTs). The functions are declared in the header file gsl_dht.h.


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



Workspace for computing discrete Hankel transforms

gsl_dht * gsl_dht_alloc(size_t size)

This function allocates a Discrete Hankel transform object of size size.

int gsl_dht_init(gsl_dht * t, double nu, double xmax)

This function initializes the transform t for the given values of nu and xmax.

gsl_dht * gsl_dht_new(size_t size, double nu, double xmax)

This function allocates a Discrete Hankel transform object of size size and initializes it for the given values of nu and xmax.

void gsl_dht_free(gsl_dht * t)

This function frees the transform t.

int gsl_dht_apply(const gsl_dht * t, double * f_in, double * f_out)

This function applies the transform t to the array f_in whose size is equal to the size of the transform. The result is stored in the array f_out which must be of the same length.

Applying this function to its output gives the original data multiplied by (X^2/j_{\nu,M})^2, up to numerical errors.

double gsl_dht_x_sample(const gsl_dht * t, int n)

This function returns the value of the n-th sample point in the unit interval, {({j_{\nu,n+1}} / {j_{\nu,M}}}) X. These are the points where the function f(t) is assumed to be sampled.

double gsl_dht_k_sample(const gsl_dht * t, int n)

This function returns the value of the n-th sample point in “k-space”, {{j_{\nu,n+1}} / X}.

References and Further Reading

The algorithms used by these functions are described in the following papers,

    1. Fisk Johnson, Comp.: Phys.: Comm.: 43, 181 (1987).
    1. Lemoine, J. Chem.: Phys.: 101, 3936 (1994).