What are the eigenfunctions of the 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.

 

[Tschew]

Hi, this is a very late reply but I found your post as one of the first results in a google search and thought other people searching would find the following link interesting:

Fourier Analysis in Polar and Spherical Coordinates

http://lmb.informatik.uni-freiburg.de/papers/download/wa_report01_08.pdf

 

[Eruvaer]

Hi, this is a very late reply but I found your post as one of the first results in a google search and thought other people searching would find the following link interesting:

Fourier Analysis in Polar and Spherical Coordinates

http://lmb.informatik.uni-freiburg.de/papers/download/wa_report01_08.pdf

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.