Fourier transform of a radial function Consider a function $f \in L^2(\mathbb{R}^n)$ such that $f$ is radial. My question is, is the Fourier transform $\hat{f}(\xi)$ automatically radial (I can see it is even in each variable $x_i$), or we need some conditions on $f$? Thanks for your help.

It indeed will be radial. It is called a Hankel transform:


Where $J_{\nu}$ is a bessel function of the first kind such that $\nu>-1/2$. For it to match fourier transform you use $J_0$ in 2 dimensional radial functions. For three dimensional radial functions, you use spherical bessel functions instead. https://en.wikipedia.org/wiki/Hankel_transform
There is a nice table on wiki. It is quite difficult to do this transform.
If you are interested in how fourier transform connects to Hankel transform, the 4th part derives it (quite short): http://math.arizona.edu/~faris/methodsweb/hankel.pdf
