Assume $f(r)=\delta(r-R)$ where $\delta(\cdot)$ is a ring delta function. In other word, $f$ is a circular delta function on a circle with radius $R$. I want to do the convolution of $f$ with itself ($f*f$).
How can I do convolution in polar(spherical) coordinates. I read that I can use Hankel and inverse Fourier transforms to obtain the convolution, however I am not into the subject and I will appreciate if some could help me with this.
Although convolutions of distributions are not defined in general, you can do it in this case. Let $$ f_\epsilon(x) = \epsilon^{-1} I_{|x| \in [R, R+\epsilon]} .$$ Then compute $f_\epsilon*f_\epsilon$, and then let $\epsilon \to 0$.
What you will see is that $f*f(x)$ is the "area" of the intersection of the boundaries of the circles of radius $R$, one centered at $0$, and the other centered at $x$. Obviously this area isn't properly defined, but you could think of thickening the boundaries of the circles so that their thicknesses are $\epsilon$, computing the area of this intersection, dividing by $\epsilon^2$, and then letting $\epsilon \to 0$. You will realize that the answer depends only on the angle at which the two circles intersect each other.