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
zeroes 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.
Specifically, let f(t) be a function on the unit interval.
Then the finite \nu-Hankel transform of f(t) is defined
to be the set of numbers g_m given by
Suppose that f is band-limited in the sense that
g_m=0 for m > M. Then we have the following
fundamental sampling theorem.
It is this discrete expression which defines the discrete Hankel
transform. 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 a scaling of
the fundamental interval, for convenience, so that one take assume the
function is defined on the interval [0,X], rather than the unit
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.