Hankel transformation of a dot product of two functions and of convolution of two functions

Question

Does a dot product of two functions in the radial domain become a convolution after using Hankel transformation on them and vice versa? I am assuming the zero order of the Bessel function is used for the Hankel transformation.

Mathematically this can be written as:

$$ \mathcal{H_0}\big\{f(r)\cdot g(r)\big\}(k)=F(k)*G(k) $$ $$ \mathcal{H_0}\big\{f(r)* g(r)\big\}(k)=F(k)\cdot G(k) $$

I know this property exists for the Laplace and Fourier transformations, but I didn't find in literature that this is a property of the Hankel transformation.

Answer 1

Since zeroth-order Hankel transform is essentially 2d Fourier transform of a radially symmetric function, a product of two functions becomes their 2d convolution after the application of Hankel transform. When written in terms of just the radial coordinates, this 2d convolution doesn't become the usual 1d convolution. Instead, it involves more complicated expressions like $\int_0^{\infty} J_0(k r_1) J_0(k r_2) J_0(k r_3) k dk$. This is discussed in the article "Operational and convolution properties of two-dimensional Fourier transforms in polar coordinates".