I want to prove that the function $J(y) = \int_{S^{n-1}} e^{i x \cdot y} d\sigma_{n-1}(x)$ is radial (i.e. $J(y) = j(|y|)$ for some $j$) where $\sigma_{n-1}$ is the surface measure on the unit sphere in n dimensions $S^{n-1}$. I've been working on this for a good while, but I can't figure out how I should do this. I'm working from Folland, so I only have the above definition of radial to work from.
As we have for any (Borel, Lebesgue) measurable function $f$ on $\mathbb{R}^n$, if $f \in L^+$ or $L^1$ then $$\int_{\mathbb{R}^n}f(x)dx = \int_{0}^{\infty} \int_{S_{n-1}}f(rx')r^{n-1}d\sigma(x')dr $$
where $r = |x|$ and $x' = x/|x|$, I thought I could make use of the fact that $r=1$ to rewrite $J(y)$ into an integral over $\mathbb{R^n}$, but I can't get much further than that. Any help is appreciated.
Note: the original problem just has $xy$ in the definition of $J(y)$, but I'm assuming that it's taking the inner product.