I'm studying some analysis and I've stumbled upon radial functions and found them useful in multiple places, but what I would really find useful is knowing if they are dense in $L^p$ spaces or in particular $L^2(R)$. If not, I would like to know why, since much stranger functions are dense in such spaces. Also what kind of space would we need in order to have them dense in it? Thanks in advance.
Edit: I know we might have problems with radial functions not being integrable, but if we could have some dense set in radial functions intersection with lp-functions that would be more what I was aiming for here.