Define the Schwartz space $\mathcal{S}\left(\mathbb{R}^{n}\right)$ and the space of tempered distributions $\mathcal{S}^{\prime}\left(\mathbb{R}^{n}\right)$. State the Fourier inversion theorem for the Fourier transform of a Schwartz function.

Consider the initial value problem:

$$\begin{gathered} \frac{\partial^{2} u}{\partial t^{2}}-\Delta u+u=0, x \in \mathbb{R}^{n}, 0<t<\infty \\ u(0, x)=f(x), \quad \frac{\partial u}{\partial t}(0, x)=0 \end{gathered}$$

for $f$ in the Schwartz space $\mathcal{S}\left(\mathbb{R}^{n}\right)$.

Show that the solution can be written as

$$u(t, x)=(2 \pi)^{-n / 2} \int_{\mathbb{R}^{n}} e^{i x \cdot \xi} \hat{u}(t, \xi) d \xi,$$

where

$$\hat{u}(t, \xi)=\cos \left(t \sqrt{1+|\xi|^{2}}\right) \hat{f}(\xi)$$

and

$$\hat{f}(\xi)=(2 \pi)^{-n / 2} \int_{\mathbb{R}^{n}} e^{-i x \cdot \xi} f(x) d x .$$

State the Plancherel-Parseval theorem and hence deduce that

$$\int_{\mathbb{R}^{n}}|u(t, x)|^{2} d x \leq \int_{\mathbb{R}^{n}}|f(x)|^{2} d x .$$