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This chapter describes functions for performing Discrete Hankel Transforms (DHTs). 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 token 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
so that
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_transform
object encapsulates this data.
The allocation function gsl_dht_transform_new
returns
a properly initialized
object which can then be used to perform transforms on data
sample vectors, for fixed \nu and M, using the
gsl_dht_transform_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 interval.
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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