I know that a function $f: \mathbb{R}^n \to \mathbb{R}$ is spherically symmetric if it is invariant under the action of an orthogonal transformation. That is, $f(Ox)=f(x)$, where $O$ is any orthogonal matrix. Now, my questions are:
Given $f$ spherically symmetric function in $\mathbb{R}^n$, is its Fourier Transform spherically symmetric, as well?
Given $f$ and $g$ spherically symmetric functions in $\mathbb{R}^n$, is the product between $f$ and $g$ spherically symmetric, as well?
The answer to (1) is yes; this is because if $O$ is orthogonal then so is $O^T$, and $O^T$ is hence measure-preserving::
$$\hat f(O\xi)=\int f(x)e^{ix\cdot(O\xi)}\,dx=\int f(x)e^{i(O^Tx)\cdot\xi}\,dx =\int f((O^T)^{-1}x)e^{ix\cdot\xi} =\int f(x)e^{ix\cdot\xi}=\hat f(\xi).$$
The second question is much easier.