(i) Define the Fourier transform $\hat{f}=\mathcal{F}(f)$ of a Schwartz function $f \in \mathcal{S}\left(\mathbb{R}^{n}\right)$, and also of a tempered distribution $u \in \mathcal{S}^{\prime}\left(\mathbb{R}^{n}\right)$.

(ii) From your definition, compute the Fourier transform of the distribution $W_{t} \in \mathcal{S}^{\prime}\left(\mathbb{R}^{3}\right)$ given by

$$W_{t}(\psi)=<W_{t}, \psi>=\frac{1}{4 \pi t} \int_{\|y\|=t} \psi(y) d \Sigma(y)$$

for every Schwartz function $\psi \in \mathcal{S}\left(\mathbb{R}^{3}\right)$. Here $d \Sigma(y)=t^{2} d \Omega(y)$ is the integration element on the sphere of radius $t$.

Hence deduce the formula of Kirchoff for the solution of the initial value problem for the wave equation in three space dimensions,

$$\frac{\partial^{2} u}{\partial t^{2}}-\Delta u=0$$

with initial data $u(0, x)=0$ and $\frac{\partial u}{\partial t}(0, x)=g(x), x \in \mathbb{R}^{3}$, where $g \in \mathcal{S}\left(\mathbb{R}^{3}\right)$. Explain briefly why the formula is also valid for arbitrary smooth $g \in C^{\infty}\left(\mathbb{R}^{3}\right)$.

(iii) Show that any $C^{2}$ solution of the initial value problem in (ii) is given by the formula derived in (ii) (uniqueness).

(iv) Show that any two $C^{2}$ solutions of the initial value problem for

$$\frac{\partial^{2} u}{\partial t^{2}}+\frac{\partial u}{\partial t}-\Delta u=0$$

with the same initial data as in (ii), also agree for any $t>0$.