Science Fair Project Encyclopedia
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 gm given by
Suppose that f is band-limited in the sense that gm = 0 for m > M. Then we have the following fundamental sampling theorem:
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.
The contents of this article is licensed from www.wikipedia.org under the GNU Free Documentation License. Click here to see the transparent copy and copyright details