I am trying to build up intuition about what the fourier transform of a spherical shell will look like but I can't say I'm making much progress.
I've also tried to dumb down the problem in 2D and consider a circle (not a disc).
ie what is the fourier transform of:
$ f = \delta(x, y) \, \forall \, x^2 + y^2 = 1 \\ f = 0, \text{otherwise} $
And in 3D
$ f = \delta(x, y, z) \, \forall\, x^2 + y^2 + z^2 = 1 \\ f = 0, \text{otherwise} $