Past exam of the mathematics course of the University of Cambridge / 2009 / ii / Paper 3 / 30B / b / Solution

Use the Fourier transform convention $\hat{\varphi }(\xi )={\int }_{{\mathbb{R}}^n}{e}^{-ix\cdot \xi }\varphi (x)dx$. For every orthogonal matrix $Q$, changing variables $x=Qy$ givessince the Jacobian has absolute determinant one and a radial function satisfies $\varphi (Qy)=\varphi (y)$. Any two vectors of the same length are related by an orthogonal transformation, so $\boxed{\hat{\varphi }\text{ is radial}}$.