Past exam of the mathematics course of the University of Cambridge / 2020 / ib / Paper 2 / 4B / Solution

Use the angular-frequency Fourier transformThenwith the continuous value $\hat{f}(0)=2A$.

The convolution $(f\ast f)(x)$ is $A^2$ times the length of the overlap of the intervals $[-1,1]$ and $[x-1,x+1]$. Thus

The convolution theorem states, for this convention, thatSince $g=(B/A^2)(f\ast f)$, it follows thatagain interpreted continuously at $k=0$, where $\hat{g}(0)=4B$.