Past exam of the mathematics course of the University of Cambridge / 2023 / iii / Paper 105 / 3 / d / Solution

For radial data, seton $\mathbb{R}$. The conditions at the origin say exactly that these are smooth odd compactly supported functions. The radial reduction $w(t,r)=ru(t,r)$ satisfies the one-dimensional wave equation, and the D'Alembert formula givesIn null coordinates this isThe limits are thereforeThey are smooth and compactly supported because $f_0,f_1$ are odd.

WriteThen $A$ is an arbitrary odd test function and $B$ is an arbitrary even test function. Conversely, every odd $A\in C_c^{\infty }(\mathbb{R})$ gives $f_0=2A$, and every even $B\in C_c^{\infty }(\mathbb{R})$ gives $f_1=-2B^{\prime }$. Thus both maps are injective andSincethe radial scattering map is