Past exam of the mathematics course of the University of Cambridge / 2007 / ii / Paper 2 / 30A / i / Solution

Use the Fourier convention $\hat{f}(\xi )={\int }_{{\mathbb{R}}^3}{e}^{-ix\cdot \xi }f(x)dx$, with inverse $(2\pi {)}^{-3}\int e^{ix\cdot \xi }\hat{f}(\xi )d\xi$. For a tempered distribution $T$, define its Fourier transform by duality:The Fourier transform maps the Schwartz space continuously onto itself, making the right side a continuous linear functional there. The use of the same negative exponential in the test-function transform is consistent with bilinear distributional pairing; other unitary or sign conventions change the normalization accordingly.