What is the 3D fourier transform of a spherical shell?

Question

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} $

Answer 1

Let $\sigma$ be the normalized arc-length measure on the circle $\mathbb T$. The Fourier transform $\mathcal F(\sigma)$ of $\sigma$ makes sense.... $$ \mathcal F(\sigma)(s,t) = \int_{\mathbb T} e^{-2\pi i(sx+ty)} \;d\sigma(x,y) = \frac{1}{2\pi} \int_0^{2\pi}e^{-2\pi i (s\cos\theta+t\sin\theta)}\,d\theta =J_0(2\pi\sqrt{s^2+t^2 }\;) $$ Here $J_0$ is a Bessel function.

More on Fourier transform of a measure: LINK

Of course if you know nothing about the theory of measure and integration, this will not mean anything to you.

Answer 2

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.