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Spherical Fourier transform

  1. Dec 6, 2007 #1
    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. jcsd
  3. Dec 6, 2007 #2


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  4. Dec 6, 2007 #3


    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?
  5. Dec 6, 2007 #4


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    In that case your equation should reduce to an ordinary differential equation in [itex]\rho[/itex] and, if I remember correctly, for the Laplace operator, at least, it is an "Euler type" equation with powers of [itex]\rho[/itex] as solution.
  6. Dec 6, 2007 #5
    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.
  7. Dec 6, 2007 #6

    Ben Niehoff

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

    [tex]\frac{A}{r} \sin kr + \frac{B}{r} \cos kr[/tex]

    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
  8. Dec 6, 2007 #7
    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.
  9. Jan 7, 2010 #8
  10. Dec 7, 2011 #9
    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.
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