Fourier transform of 3D Sinc function

Question

What is the Fourier trasnform of the function

$$\frac{\sin(P|\mathbf{x-y}|)}{|\mathbf{x-y}|}$$

where $P$ is a real parameter and $\mathbf{y}$ is a fixed point in three-dimensional space?

Answer 1

It is a function with support on a sphere of radius $P$: $$ \frac {e^{i(\xi,y)}}{4\pi P}\delta_{S_P}(\xi). $$ It is closely related to the formula for the fundamental solution of the wave equation in $\mathbb R^3$. Namely, if to do a Fourier transform on the space variables of the equation $$ u_{PP}-\Delta u=\delta(x,P), $$ then solving the resulting ODE one gets almost the function in question, $\theta(P)\frac{\sin(P|\mathbf{\xi}|)}{|\mathbf{\xi}|}$. Here $\theta$ is the Heaviside step function.

Answer 2

I found it is

$i\sqrt{\frac{\pi}{2}}\frac{e^{-i\mathbf{k}\cdot\mathbf{y}}}{|\mathbf{k}|} \delta(P-|\mathbf{k}|)$