TITLE: What is the 3D fourier transform of a spherical shell?
QUESTION [5 upvotes]: 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}
$

REPLY [3 votes]: In 3D: $\frac{\sin(kr)}{kr}$
Source:
Vembu, S. "Fourier transformation of the n-dimensional radial delta function." The Quarterly Journal of Mathematics 12.1 (1961): 165-168.