
# 5.E: Fourier Transform (Exercises)

### Q5.1

Show

$$\int_{{\mathbb{R}}^n}{\rm e}^{-|y|^2/2}\ dy=(2\pi)^{n/2}.$$

### Q5.2

Show that $$u\in \mathcal{S}(\mathbb{R}n^)$$ implies $$\hat{u},\ \widetilde{u}\in\mathcal{S}(\mathbb{R}^n)$$.

### Q5.3

Give examples for functions $$p(x,\xi)$$ which satisfy $$p(x,\xi)\in S^m$$.

### Q5.4

Find a formal solution of Cauchy's initial value problem  for the wave equation by using Fourier's transform.

### Contributors:

• Integrated by Justin Marshall.