# Spherical Fourier transform

1. Dec 6, 2007

### 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?

2. Dec 6, 2007

### HallsofIvy

Staff Emeritus
3. Dec 6, 2007

### christianjb

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?

4. Dec 6, 2007

### HallsofIvy

Staff Emeritus
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.

5. Dec 6, 2007

### 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.

6. Dec 6, 2007

### 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.

Last edited: Dec 6, 2007
7. Dec 6, 2007

### 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.

8. Jan 7, 2010

9. Dec 7, 2011

### Eruvaer

Thanks a LOT, dude, this paper just really helped me! Good thing that the internet doesn't forget so even such old threads can be helpful.