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Discrete Hankel TransformsThis chapter describes functions for performing Discrete Hankel Transforms (DHTs). The functions are declared in the header file `gsl_dht.h'. DefinitionsThe 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 \nuHankel transform of f(t) is defined to be the set of numbers g_m given by so that Suppose that f is bandlimited 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
\nuHankel transform of size M1. The coefficients of
this matrix, being dependent on \nu and M, must be
precomputed and stored; the 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. Functions
References and Further ReadingThe algorithms used by these functions are described in the following papers,
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