Spherical Fourier transform

[christianjb]

Does anyone know what the eigenfunctions of the spherical Fourier transform are? I want to expand a spherically symmetric function in these eigenfunctions.

Are they Bessel functions? Legendre functions?

[HallsofIvy]

They are the "spherical harmonics". Yes, they involve teh Legendre functions. Check this: http://en.wikipedia.org/wiki/Spherical_harmonics#Spherical_harmonics_expansion.

[christianjb]

They are the "spherical harmonics". Yes, they involve teh Legendre functions. Check this: http://en.wikipedia.org/wiki/Spherical_harmonics#Spherical_harmonics_expansion.


Thanks.

I have a spherically symmetric function - i.e. no theta/phi dependence. The spherical harmonics account for only the theta/phi dependence- or am I missing something?

[HallsofIvy]

In that case your equation should reduce to an ordinary differential equation in \rho and, if I remember correctly, for the Laplace operator, at least, it is an "Euler type" equation with powers of \rho as solution.

[christianjb]

I'm not solving a differential eqn. I'm looking for an orthogonal basis where each basis function is an eigenfunction of the spherical Fourier transform.

[Ben Niehoff]

I'm pretty sure you can still expand it in terms of A \sin kr + B \cos kr. It is, after all, some function of r, so you can just Fourier-transform it normally. If not, then try

\frac{A}{r} \sin kr + \frac{B}{r} \cos kr

This is a solution of the wave equation in 3 dimensions, in the same since that sin and cos are solutions in 1 dimension, and Bessel functions are solutions in 2 dimensions.

I could be totally wrong here, though.

Yet another option is to write down the 3D Fourier transform in Cartesian coordinates, and transform it to spherical coordinates.

[christianjb]

Thanks- but I'm specifically looking for the eigenfunctions of the spherically symmetric Fourier transform. I've already got numerical solutions for the spherical FT.