Past exam of the mathematics course of the University of Cambridge / 2017 / ii / Paper 2 / 24J / b / Solution

Here Fourier inversion means the absolute-integral version: $\hat{f}\in L^1({\mathbb{R}}^d)$ andThis convention is necessary for the bounded continuous version in part (d). Conditional inversion by symmetric truncation is a different statement and need not have that conclusion.