(a) State the Fourier inversion theorem for Schwartz functions $\mathcal{S}(\mathbb{R})$ on the real line. Define the Fourier transform of a tempered distribution and compute the Fourier transform of the distribution defined by the function $F(x)=\frac{1}{2}$ for $-t \leqslant x \leqslant+t$ and $F(x)=0$ otherwise. (Here $t$ is any positive number.)

Use the Fourier transform in the $x$ variable to deduce a formula for the solution to the one dimensional wave equation

$$u_{t t}-u_{x x}=0, \quad \text { with initial data } \quad u(0, x)=0, \quad u_{t}(0, x)=g(x),$$

for $g$ a Schwartz function. Explain what is meant by "finite propagation speed" and briefly explain why the formula you have derived is in fact valid for arbitrary smooth $g \in C^{\infty}(\mathbb{R})$.

(b) State a theorem on the representation of a smooth $2 \pi$-periodic function $g$ as a Fourier series

$$g(x)=\sum_{\alpha \in \mathbb{Z}} \hat{g}(\alpha) e^{i \alpha x}$$

and derive a representation for solutions to $(*)$ as Fourier series in $x$.

(c) Verify that the formulae obtained in (a) and (b) agree for the case of smooth $2 \pi$ periodic $g$.