Past exam of the mathematics course of the University of Cambridge / 2006 / ib / Paper 4 / 15D / a / Solution

For example, it suffices to assume $f,g\in L^1(\mathbb{R})$. Indeed the absolute double integral isThus Fubini's theorem permits interchange in the Fourier transform of the convolution. With $u=x-t$,Hence the convolution theorem in this normalization isThere is no extra $2\pi$ in this formula; that factor belongs to the inverse transform and Parseval identity for the convention being used.