discrete hankel transform

In mathematics and statistics, 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. Likewise, the discrete Hankel transform is related to the continuous Hankel transform just as the discrete Fourier transform is related to the continuous Fourier transform. Specifically, let f(t) be a function on the unit interval. Then the finite ν-Hankel transform of f(t) is defined to be the set of numbers g_m given by

g_m = \int_0^1 t J_\nu(j_{\nu,m} t)f(t)\,dt

so that

f(t) = \sum^\infty_{m=1} \frac{2J_\nu(j_{\nu,m} x)}{J_{\nu+1}(j_{\nu,m})^2} g_m.

Suppose that f is band-limited in the sense that g_m = 0 for m > M. Then we have the following fundamental sampling theorem:

g_m = \frac{2}{j^2_{\nu,M}} \sum_{k=1}^{M-1} f(\frac{j_{\nu,k}}{j_{\nu,M}}) \frac{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. The kernel in the summation above defines the matrix of the ν-Hankel transform of size M - 1. 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.

Discrete Hankel Transform

Further reading